Mathematics · Textbook solutions

Limits

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 181 questions

7.1.2 Definition of a Limit

3 q

Solved Examples

Worked · 3
  1. 7.1.2 SolvedEx.1
    Consider the example f(x)=3x+1f(x) = 3x + 1, take a=0a = 0 and l=1l = 1. Verify the ε\varepsilon-δ\delta definition of the limit, i.e. find δ>0\delta > 0 such that 0<∣x−0∣<δ0 < |x - 0| < \delta implies ∣f(x)−l∣<ε|f(x) - l| < \varepsilon.
  2. 7.1.2 SolvedEx.2
    f(x)=x2f(x) = x^2. Here take a=3a = 3 and l=9l = 9. Find δ>0\delta > 0 such that 0<∣x−3∣<δ0 < |x - 3| < \delta implies ∣x2−9∣<ε|x^2 - 9| < \varepsilon.
  3. 7.1.2 SolvedEx.3
    f(x)=[x]f(x) = [x], 2<x<42 < x < 4, where [x][x] is the greatest integer function. Study the limits of f(x)f(x) as x→3x \to 3 and as x→2.7x \to 2.7.

7.1.6 Existence of a Limit

1 q

Worked Example

Worked · 1
  1. 7.1.6 SolvedEx.1
    Find left hand limit and right hand limit for the following function. f(x)={3x+1if x<17x2−3if x≥1f(x) = \begin{cases} 3x + 1 & \text{if } x < 1 \\ 7x^2 - 3 & \text{if } x \ge 1 \end{cases}

7.1.7 Algebra of Limits

6 q

Solved Examples

Worked · 6
  1. 7.1.7 SolvedEx.1
    Evaluate lim⁡n→3(∑r=1nr2)\lim_{n \to 3} \left( \sum_{r=1}^{n} r^2 \right)
  2. 7.1.7 SolvedEx.2
    Evaluate lim⁡y→2[(y2−3)(y+2)]\lim_{y \to 2} \left[ (y^2 - 3)(y + 2) \right]
  3. 7.1.7 SolvedEx.3
    Evaluate lim⁡x→3(6+x−7−xx)\lim_{x \to 3} \left( \dfrac{\sqrt{6 + x} - \sqrt{7 - x}}{x} \right)
  4. 7.1.7 SolvedEx.4
    Evaluate lim⁡x→1(1x−1x−1)\lim_{x \to 1} \left( \dfrac{\dfrac{1}{x} - 1}{x - 1} \right)
  5. 7.1.7 SolvedEx.5
    Discuss the limit of the following function as xx tends to 3 if f(x)={x2+x+1,2≤x≤32x+1,3<x≤4f(x) = \begin{cases} x^2 + x + 1, & 2 \le x \le 3 \\ 2x + 1, & 3 < x \le 4 \end{cases}
  6. 7.1.7 SolvedEx.6
    For a given ε>0\varepsilon > 0, find δ>0\delta > 0 such that whenever ∣x−a∣<δ|x - a| < \delta, we have ∣f(x)−l∣<ε|f(x) - l| < \varepsilon so that lim⁡x→1(4x+3)=7\lim_{x \to 1} (4x + 3) = 7

7.1.8 Limit Theorem

4 q

Solved Examples

Worked · 4
  1. 7.1.8 SolvedEx.1
    Evaluate lim⁡x→5(x4−625x−5)\lim_{x \to 5} \left( \dfrac{x^4 - 625}{x - 5} \right)
  2. 7.1.8 SolvedEx.2
    Evaluate lim⁡x→2(x7−128x5−32)\lim_{x \to 2} \left( \dfrac{x^7 - 128}{x^5 - 32} \right)
  3. 7.1.8 SolvedEx.3
    If lim⁡x→4[xn−4nx−4]=48\lim_{x \to 4} \left[ \dfrac{x^n - 4^n}{x - 4} \right] = 48 and n∈Nn \in N, find nn.
  4. 7.1.8 SolvedEx.4
    Evaluate lim⁡x→1[2x−226+x3−3]\lim_{x \to 1} \left[ \dfrac{2x - 2}{\sqrt[3]{26 + x} - 3} \right]

Exercise 7.1

20 q
  1. Evaluate the following limits :
    Ex 7.1 I Q1
    lim⁡z→−3[z+6z]\lim_{z \to -3} \left[ \dfrac{\sqrt{z + 6}}{z} \right]
  2. Ex 7.1 I Q2
    lim⁡y→−3[y5+243y3+27]\lim_{y \to -3} \left[ \dfrac{y^5 + 243}{y^3 + 27} \right]
  3. Ex 7.1 I Q3
    lim⁡z→−5[(1z+15)z+5]\lim_{z \to -5} \left[ \dfrac{\left( \dfrac{1}{z} + \dfrac{1}{5} \right)}{z + 5} \right]
  4. Evaluate the following limits :
    Ex 7.1 II Q1
    lim⁡x→3[2x+6x]\lim_{x \to 3} \left[ \dfrac{\sqrt{2x + 6}}{x} \right]
  5. Ex 7.1 II Q2
    lim⁡x→2[x−3−2−3x−2]\lim_{x \to 2} \left[ \dfrac{x^{-3} - 2^{-3}}{x - 2} \right]
  6. Ex 7.1 II Q3
    lim⁡x→5[x3−125x5−3125]\lim_{x \to 5} \left[ \dfrac{x^3 - 125}{x^5 - 3125} \right]
  7. Ex 7.1 II Q4
    If lim⁡x→1[x4−1x−1]=lim⁡x→a[x3−a3x−a]\lim_{x \to 1} \left[ \dfrac{x^4 - 1}{x - 1} \right] = \lim_{x \to a} \left[ \dfrac{x^3 - a^3}{x - a} \right], find all possible values of aa.
  8. Evaluate the following limits :
    Ex 7.1 III Q1
    lim⁡x→1[x+x2+x3+………+xn−nx−1]\lim_{x \to 1} \left[ \dfrac{x + x^2 + x^3 + \ldots\ldots\ldots + x^n - n}{x - 1} \right]
  9. Ex 7.1 III Q2
    lim⁡x→7[(x3−73)(x3+73)x−7]\lim_{x \to 7} \left[ \dfrac{\left( \sqrt[3]{x} - \sqrt[3]{7} \right)\left( \sqrt[3]{x} + \sqrt[3]{7} \right)}{x - 7} \right]
  10. Ex 7.1 III Q3
    If lim⁡x→5[xk−5kx−5]=500\lim_{x \to 5} \left[ \dfrac{x^k - 5^k}{x - 5} \right] = 500, find all possible values of kk.
  11. Ex 7.1 III Q4
    lim⁡x→0[(1−x)8−1(1−x)2−1]\lim_{x \to 0} \left[ \dfrac{(1 - x)^8 - 1}{(1 - x)^2 - 1} \right]
  12. Ex 7.1 III Q5
    lim⁡x→0[1+x3−1+xx]\lim_{x \to 0} \left[ \dfrac{\sqrt[3]{1 + x} - \sqrt{1 + x}}{x} \right]
  13. Ex 7.1 III Q6
    lim⁡y→1[2y−27+y3−2]\lim_{y \to 1} \left[ \dfrac{2y - 2}{\sqrt[3]{7 + y} - 2} \right]
  14. Ex 7.1 III Q7
    lim⁡z→a[(z+2)32−(a+2)32z−a]\lim_{z \to a} \left[ \dfrac{(z + 2)^{\frac{3}{2}} - (a + 2)^{\frac{3}{2}}}{z - a} \right]
  15. Ex 7.1 III Q8
    lim⁡x→7[x3−343x−7]\lim_{x \to 7} \left[ \dfrac{x^3 - 343}{\sqrt{x} - \sqrt{7}} \right]
  16. Ex 7.1 III Q9
    lim⁡x→1(x+x3+x5+…+x2n−1−nx−1)\lim_{x \to 1} \left( \dfrac{x + x^3 + x^5 + \ldots + x^{2n-1} - n}{x - 1} \right)
  17. In the following examples, given ε>0\varepsilon > 0, find a δ>0\delta > 0 such that whenever ∣x−a∣<δ|x - a| < \delta, we must have ∣f(x)−l∣<ε|f(x) - l| < \varepsilon
    Ex 7.1 IV Q1
    lim⁡x→2(2x+3)=7\lim_{x \to 2} (2x + 3) = 7
  18. Ex 7.1 IV Q2
    lim⁡x→−3(3x+2)=−7\lim_{x \to -3} (3x + 2) = -7
  19. Ex 7.1 IV Q3
    lim⁡x→2(x2−1)=3\lim_{x \to 2} (x^2 - 1) = 3
  20. Ex 7.1 IV Q4
    lim⁡x→1(x2+x+1)=3\lim_{x \to 1} (x^2 + x + 1) = 3

7.2 Method of Factorization

5 q

Solved Examples

Worked · 5
  1. 7.2 SolvedEx.1
    Evaluate lim⁡z→3[z(2z−3)−9z2−4z+3]\lim_{z \to 3}\left[\dfrac{z(2z-3)-9}{z^{2}-4z+3}\right]
  2. 7.2 SolvedEx.2
    Evaluate lim⁡x→4[(x3−8x2+16x)9(x2−x−12)18]\lim_{x \to 4}\left[\dfrac{\left(x^{3}-8x^{2}+16x\right)^{9}}{\left(x^{2}-x-12\right)^{18}}\right]
  3. 7.2 SolvedEx.3
    Evaluate lim⁡x→1[1x−1+21−x2]\lim_{x \to 1}\left[\dfrac{1}{x-1}+\dfrac{2}{1-x^{2}}\right]
  4. 7.2 SolvedEx.4
    Evaluate lim⁡x→1[x3+x2−5x+3x2−1]\lim_{x \to 1}\left[\dfrac{x^{3}+x^{2}-5x+3}{x^{2}-1}\right]
  5. 7.2 SolvedEx.5
    Evaluate lim⁡x→0(1−x2+13x2)\lim_{x \to 0}\left(\dfrac{1-\sqrt[3]{x^{2}+1}}{x^{2}}\right)

Exercise 7.2

16 q
  1. Evaluate the following limits :
    Ex 7.2 I Q1
    lim⁡z→2[z2−5z+6z2−4]\lim_{z \to 2}\left[\dfrac{z^{2}-5z+6}{z^{2}-4}\right]
  2. Ex 7.2 I Q2
    lim⁡x→−3[x+3x2+4x+3]\lim_{x \to -3}\left[\dfrac{x+3}{x^{2}+4x+3}\right]
  3. Ex 7.2 I Q3
    lim⁡y→0[5y3+8y23y4−16y2]\lim_{y \to 0}\left[\dfrac{5y^{3}+8y^{2}}{3y^{4}-16y^{2}}\right]
  4. Ex 7.2 I Q4
    lim⁡x→−2[−2x−4x3+2x2]\lim_{x \to -2}\left[\dfrac{-2x-4}{x^{3}+2x^{2}}\right]
  5. Ex 7.2 I Q5
    lim⁡x→3[x2+2x−15x2−5x+6]\lim_{x \to 3}\left[\dfrac{x^{2}+2x-15}{x^{2}-5x+6}\right]
  6. Evaluate the following limits :
    Ex 7.2 II Q1
    lim⁡u→1[u4−1u3−1]\lim_{u \to 1}\left[\dfrac{u^{4}-1}{u^{3}-1}\right]
  7. Ex 7.2 II Q2
    lim⁡x→3[1x−3−9xx3−27]\lim_{x \to 3}\left[\dfrac{1}{x-3}-\dfrac{9x}{x^{3}-27}\right]
  8. Ex 7.2 II Q3
    lim⁡x→2[x3−4x2+4xx2−1]\lim_{x \to 2}\left[\dfrac{x^{3}-4x^{2}+4x}{x^{2}-1}\right]
  9. Ex 7.2 II Q4
    lim⁡Δx→0[(x+Δx)2−2(x+Δx)+1−(x2−2x+1)Δx]\lim_{\Delta x \to 0}\left[\dfrac{(x+\Delta x)^{2}-2(x+\Delta x)+1-\left(x^{2}-2x+1\right)}{\Delta x}\right]
  10. Ex 7.2 II Q5
    lim⁡x→2[x2+x2−4x2−3x2+4]\lim_{x \to \sqrt{2}}\left[\dfrac{x^{2}+x\sqrt{2}-4}{x^{2}-3x\sqrt{2}+4}\right]
  11. Ex 7.2 II Q6
    lim⁡x→2[x3−7x+6x3−7x2+16x−12]\lim_{x \to 2}\left[\dfrac{x^{3}-7x+6}{x^{3}-7x^{2}+16x-12}\right]
  12. Evaluate the Following limits :
    Ex 7.2 III Q1
    lim⁡y→12[1−8y3y−4y3]\lim_{y \to \frac{1}{2}}\left[\dfrac{1-8y^{3}}{y-4y^{3}}\right]
  13. Ex 7.2 III Q2
    lim⁡x→1[x−2x2−x−1x3−3x2+2x]\lim_{x \to 1}\left[\dfrac{x-2}{x^{2}-x}-\dfrac{1}{x^{3}-3x^{2}+2x}\right]
  14. Ex 7.2 III Q3
    lim⁡x→1[x4−3x2+2x3−5x2+3x+1]\lim_{x \to 1}\left[\dfrac{x^{4}-3x^{2}+2}{x^{3}-5x^{2}+3x+1}\right]
  15. Ex 7.2 III Q4
    lim⁡x→1[x+2x2−5x+4+x−43(x2−3x+2)]\lim_{x \to 1}\left[\dfrac{x+2}{x^{2}-5x+4}+\dfrac{x-4}{3\left(x^{2}-3x+2\right)}\right]
  16. Ex 7.2 III Q5
    lim⁡x→a[1x2−3ax+2a2+12x2−3ax+a2]\lim_{x \to a}\left[\dfrac{1}{x^{2}-3ax+2a^{2}}+\dfrac{1}{2x^{2}-3ax+a^{2}}\right]

7.3 Method of Rationalization

3 q

Solved Examples

Worked · 3
  1. 7.3 SolvedEx.1
    Evaluate : lim⁡x→0(1+x−1x)\lim_{x \to 0}\left(\dfrac{\sqrt{1+x}-1}{x}\right)
  2. 7.3 SolvedEx.2
    Evaluate lim⁡z→0[(b+z)12−(b−z)12z]\lim_{z \to 0}\left[\dfrac{(b+z)^{\frac{1}{2}}-(b-z)^{\frac{1}{2}}}{z}\right]
  3. 7.3 SolvedEx.3
    Evaluate lim⁡x→4(x2+x−20x2−7−25−x2)\lim_{x \to 4}\left(\dfrac{x^{2}+x-20}{\sqrt{x^{2}-7}-\sqrt{25-x^{2}}}\right)

Exercise 7.3

14 q
  1. Evaluate the following limits :
    Ex 7.3 I Q1
    lim⁡x→0[6+x+x2−6x]\lim_{x \to 0}\left[\dfrac{\sqrt{6+x+x^{2}}-\sqrt{6}}{x}\right]
  2. Ex 7.3 I Q2
    lim⁡x→3[2x+3−4x−3x2−9]\lim_{x \to 3}\left[\dfrac{\sqrt{2x+3}-\sqrt{4x-3}}{x^{2}-9}\right]
  3. Ex 7.3 I Q3
    lim⁡y→0[1−y2−1+y2y2]\lim_{y \to 0}\left[\dfrac{\sqrt{1-y^{2}}-\sqrt{1+y^{2}}}{y^{2}}\right]
  4. Ex 7.3 I Q4
    lim⁡x→2[2+x−6−xx−2]\lim_{x \to 2}\left[\dfrac{\sqrt{2+x}-\sqrt{6-x}}{\sqrt{x}-\sqrt{2}}\right]
  5. Evaluate the following limits :
    Ex 7.3 II Q1
    lim⁡x→a[a+2x−3x3a+x−2x]\lim_{x \to a}\left[\dfrac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}\right]
  6. Ex 7.3 II Q2
    lim⁡x→2[x2−4x+2−3x−2]\lim_{x \to 2}\left[\dfrac{x^{2}-4}{\sqrt{x+2}-\sqrt{3x-2}}\right]
  7. Ex 7.3 II Q3
    lim⁡x→2[1+2+x−3x−2]\lim_{x \to 2}\left[\dfrac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2}\right]
  8. Ex 7.3 II Q4
    lim⁡y→0[a+y−aya+y]\lim_{y \to 0}\left[\dfrac{\sqrt{a+y}-\sqrt{a}}{y\sqrt{a+y}}\right]
  9. Ex 7.3 II Q5
    lim⁡x→0(x2+9−2x2+93x2+4−2x2+4)\lim_{x \to 0}\left(\dfrac{\sqrt{x^{2}+9}-\sqrt{2x^{2}+9}}{\sqrt{3x^{2}+4}-\sqrt{2x^{2}+4}}\right)
  10. Evaluate the Following limits :
    Ex 7.3 III Q1
    lim⁡x→1[x2+xx−2x−1]\lim_{x \to 1}\left[\dfrac{x^{2}+x\sqrt{x}-2}{x-1}\right]
  11. Ex 7.3 III Q2
    lim⁡x→0[1+x2−1+x1+x3−1+x]\lim_{x \to 0}\left[\dfrac{\sqrt{1+x^{2}}-\sqrt{1+x}}{\sqrt{1+x^{3}}-\sqrt{1+x}}\right]
  12. Ex 7.3 III Q3
    lim⁡x→4[x2+x−203x+4−4]\lim_{x \to 4}\left[\dfrac{x^{2}+x-20}{\sqrt{3x+4}-4}\right]
  13. Ex 7.3 III Q4
    lim⁡z→4[3−5+z1−5−z]\lim_{z \to 4}\left[\dfrac{3-\sqrt{5+z}}{1-\sqrt{5-z}}\right]
  14. Ex 7.3 III Q5
    lim⁡x→0(3x9−x−1x)\lim_{x \to 0}\left(\dfrac{3}{x\sqrt{9-x}}-\dfrac{1}{x}\right)

7.4 Limits of Trigonometric Functions

8 q

Solved Examples

Worked · 8
  1. 7.4 SolvedEx.1
    If 3x2+2≤f(x)≤5x2−63x^2 + 2 \le f(x) \le 5x^2 - 6 for all x∈Rx \in R, then find lim⁡x→−2f(x)\lim_{x \to -2} f(x).
  2. 7.4 SolvedEx.2
    Evaluate: lim⁡x→π4(cot⁡x−1cosec⁡2x−2)\lim_{x \to \frac{\pi}{4}}\left(\frac{\cot x - 1}{\operatorname{cosec}^2 x - 2}\right)
  3. 7.4 SolvedEx.3
    Evaluate: lim⁡x→3π2(3−sin⁡x−2cos⁡2x)\lim_{x \to \frac{3\pi}{2}}\left(\frac{\sqrt{3 - \sin x} - 2}{\cos^2 x}\right)
  4. 7.4 SolvedEx.4
    Evaluate: lim⁡θ→0[sin⁡7θθ]\lim_{\theta \to 0}\left[\frac{\sin 7\theta}{\theta}\right]
  5. 7.4 SolvedEx.5
    Evaluate: lim⁡x→0[sin⁡8xtan⁡4x]\lim_{x \to 0}\left[\frac{\sin 8x}{\tan 4x}\right]
  6. 7.4 SolvedEx.6
    Evaluate: lim⁡x→0[2sin⁡x−sin⁡2xx3]\lim_{x \to 0}\left[\frac{2\sin x - \sin 2x}{x^3}\right]
  7. 7.4 SolvedEx.7
    Evaluate: lim⁡x→0[sin⁡x2(1−cos⁡x2)x6]\lim_{x \to 0}\left[\frac{\sin x^2\left(1 - \cos x^2\right)}{x^6}\right]
  8. 7.4 SolvedEx.8
    Evaluate: lim⁡x→0(cos⁡5x∘−cos⁡3x∘x2)\lim_{x \to 0}\left(\frac{\cos 5x^{\circ} - \cos 3x^{\circ}}{x^2}\right)

Exercise 7.4

11 q
  1. Evaluate the following limits :
    Ex 7.4 I Q1
    lim⁡θ→0[sin⁡(mθ)tan⁡(nθ)]\lim_{\theta \to 0}\left[\frac{\sin(m\theta)}{\tan(n\theta)}\right]
  2. Ex 7.4 I Q2
    lim⁡θ→0[1−cos⁡2θθ2]\lim_{\theta \to 0}\left[\frac{1 - \cos 2\theta}{\theta^2}\right]
  3. Ex 7.4 I Q3
    lim⁡x→0[x⋅tan⁡x1−cos⁡x]\lim_{x \to 0}\left[\frac{x \cdot \tan x}{1 - \cos x}\right]
  4. Ex 7.4 I Q4
    lim⁡x→0(sec⁡x−1x2)\lim_{x \to 0}\left(\frac{\sec x - 1}{x^2}\right)
  5. Evaluate the Following limits :
    Ex 7.4 II Q1
    lim⁡x→0[1−cos⁡(nx)1−cos⁡(mx)]\lim_{x \to 0}\left[\frac{1 - \cos(nx)}{1 - \cos(mx)}\right]
  6. Ex 7.4 II Q2
    lim⁡x→π6[2−cosec⁡xcot⁡2x−3]\lim_{x \to \frac{\pi}{6}}\left[\frac{2 - \operatorname{cosec} x}{\cot^2 x - 3}\right]
  7. Ex 7.4 II Q3
    lim⁡x→π4[cos⁡x−sin⁡xcos⁡2x]\lim_{x \to \frac{\pi}{4}}\left[\frac{\cos x - \sin x}{\cos 2x}\right]
  8. Evaluate the following limits :
    Ex 7.4 III Q1
    lim⁡x→0[cos⁡(ax)−cos⁡(bx)cos⁡(cx)−1]\lim_{x \to 0}\left[\frac{\cos(ax) - \cos(bx)}{\cos(cx) - 1}\right]
  9. Ex 7.4 III Q2
    lim⁡x→π[1−cos⁡x−2sin⁡2x]\lim_{x \to \pi}\left[\frac{\sqrt{1 - \cos x} - \sqrt{2}}{\sin^2 x}\right]
  10. Ex 7.4 III Q3
    lim⁡x→π4[tan⁡2x−cot⁡2xsec⁡x−cosec⁡x]\lim_{x \to \frac{\pi}{4}}\left[\frac{\tan^2 x - \cot^2 x}{\sec x - \operatorname{cosec} x}\right]
  11. Ex 7.4 III Q4
    lim⁡x→π6[2sin⁡2x+sin⁡x−12sin⁡2x−3sin⁡x+1]\lim_{x \to \frac{\pi}{6}}\left[\frac{2\sin^2 x + \sin x - 1}{2\sin^2 x - 3\sin x + 1}\right]

7.5 Substitution Method

4 q

Solved Examples

Worked · 4
  1. 7.5 SolvedEx.1
    Evaluate: lim⁡x→π2[cos⁡xx−π2]\lim_{x \to \frac{\pi}{2}}\left[\frac{\cos x}{x - \frac{\pi}{2}}\right]
  2. 7.5 SolvedEx.2
    Evaluate: lim⁡x→a[cos⁡x−cos⁡ax−a]\lim_{x \to a}\left[\frac{\cos x - \cos a}{x - a}\right]
  3. 7.5 SolvedEx.3
    Evaluate: lim⁡x→1[1+cos⁡πx(1−x)2]\lim_{x \to 1}\left[\frac{1 + \cos \pi x}{(1 - x)^2}\right]
  4. 7.5 SolvedEx.4
    Evaluate: lim⁡x→π3[3−tan⁡xπ−3x]\lim_{x \to \frac{\pi}{3}}\left[\frac{\sqrt{3} - \tan x}{\pi - 3x}\right]

Exercise 7.5

10 q
  1. Evaluate the following
    Ex 7.5 I Q1
    lim⁡x→π2[cosec⁡x−1(π2−x)2]\lim_{x \to \frac{\pi}{2}}\left[\frac{\operatorname{cosec} x - 1}{\left(\frac{\pi}{2} - x\right)^2}\right]
  2. Ex 7.5 I Q2
    lim⁡x→asin⁡x−sin⁡ax5−a5\lim_{x \to a}\frac{\sin x - \sin a}{\sqrt[5]{x} - \sqrt[5]{a}}
  3. Ex 7.5 I Q3
    lim⁡x→π[5+cos⁡x−2(π−x)2]\lim_{x \to \pi}\left[\frac{\sqrt{5 + \cos x} - 2}{(\pi - x)^2}\right]
  4. Ex 7.5 I Q4
    lim⁡x→π6[cos⁡x−3sin⁡xπ−6x]\lim_{x \to \frac{\pi}{6}}\left[\frac{\cos x - \sqrt{3}\sin x}{\pi - 6x}\right]
  5. Ex 7.5 I Q5
    lim⁡x→1[1−x2sin⁡πx]\lim_{x \to 1}\left[\frac{1 - x^2}{\sin \pi x}\right]
  6. Evaluate the following
    Ex 7.5 II Q1
    lim⁡x→π6[2sin⁡x−1π−6x]\lim_{x \to \frac{\pi}{6}}\left[\frac{2\sin x - 1}{\pi - 6x}\right]
  7. Ex 7.5 II Q2
    lim⁡x→π4[2−cos⁡x−sin⁡x(4x−π)2]\lim_{x \to \frac{\pi}{4}}\left[\frac{\sqrt{2} - \cos x - \sin x}{(4x - \pi)^2}\right]
  8. Ex 7.5 II Q3
    lim⁡x→π6[2−3cos⁡x−sin⁡x(6x−π)2]\lim_{x \to \frac{\pi}{6}}\left[\frac{2 - \sqrt{3}\cos x - \sin x}{(6x - \pi)^2}\right]
  9. Ex 7.5 II Q4
    lim⁡x→a[sin⁡(x)−sin⁡(a)x−a]\lim_{x \to a}\left[\frac{\sin\left(\sqrt{x}\right) - \sin\left(\sqrt{a}\right)}{x - a}\right]
  10. Ex 7.5 II Q5
    lim⁡x→π2[cos⁡3x+3cos⁡x(2x−π)3]\lim_{x \to \frac{\pi}{2}}\left[\frac{\cos 3x + 3\cos x}{(2x - \pi)^3}\right]

7.6 Limits of Exponential and Logarithmic Functions

7 q

Solved Examples

Worked · 7
  1. 7.6 SolvedEx.1
    Evaluate : lim⁡x→0[5x−1sin⁡x]\lim_{x \to 0}\left[\dfrac{5^{x} - 1}{\sin x}\right]
  2. 7.6 SolvedEx.2
    Evaluate : lim⁡x→0[5x−3xx]\lim_{x \to 0}\left[\dfrac{5^{x} - 3^{x}}{x}\right]
  3. 7.6 SolvedEx.3
    Evaluate : lim⁡x→0[1+5x6]1x\lim_{x \to 0}\left[1 + \dfrac{5x}{6}\right]^{\frac{1}{x}}
  4. 7.6 SolvedEx.4
    Evaluate : lim⁡x→0[3x+22−5x]13x\lim_{x \to 0}\left[\dfrac{3x + 2}{2 - 5x}\right]^{\frac{1}{3x}}
  5. 7.6 SolvedEx.5
    Evaluate : lim⁡x→0[log⁡4+log⁡(0.25+x)x]\lim_{x \to 0}\left[\dfrac{\log 4 + \log(0.25 + x)}{x}\right]
  6. 7.6 SolvedEx.6
    Evaluate : lim⁡x→0(e2x+e−2x−2xsin⁡x)\lim_{x \to 0}\left(\dfrac{e^{2x} + e^{-2x} - 2}{x \sin x}\right)
  7. 7.6 SolvedEx.7
    Evaluate : lim⁡x→0(21x−7x−3x+1xlog⁡(1+x))\lim_{x \to 0}\left(\dfrac{21^{x} - 7^{x} - 3^{x} + 1}{x \log(1 + x)}\right)

Exercise 7.6

16 q
  1. Ex 7.6 I Q1
    Evaluate the following limit : lim⁡x→0[9x−5x4x−1]\lim_{x \to 0}\left[\dfrac{9^{x} - 5^{x}}{4^{x} - 1}\right]
  2. Ex 7.6 I Q2
    Evaluate the following limit : lim⁡x→0[5x+3x−2x−1x]\lim_{x \to 0}\left[\dfrac{5^{x} + 3^{x} - 2^{x} - 1}{x}\right]
  3. Ex 7.6 I Q3
    Evaluate the following limit : lim⁡x→0(ax+bx+cx−3sin⁡x)\lim_{x \to 0}\left(\dfrac{a^{x} + b^{x} + c^{x} - 3}{\sin x}\right)
  4. Ex 7.6 I Q4
    Evaluate the following limit : lim⁡x→0(6x+5x+4x−3x+1sin⁡x)\lim_{x \to 0}\left(\dfrac{6^{x} + 5^{x} + 4^{x} - 3^{x+1}}{\sin x}\right)
  5. Ex 7.6 I Q5
    Evaluate the following limit : lim⁡x→0(8sin⁡x−2tan⁡xe2x−1)\lim_{x \to 0}\left(\dfrac{8^{\sin x} - 2^{\tan x}}{e^{2x} - 1}\right)
  6. Ex 7.6 II Q1
    Evaluate the following limit : lim⁡x→0[3x+3−x−2x⋅tan⁡x]\lim_{x \to 0}\left[\dfrac{3^{x} + 3^{-x} - 2}{x \cdot \tan x}\right]
  7. Ex 7.6 II Q2
    Evaluate the following limit : lim⁡x→0[3+x3−x]1x\lim_{x \to 0}\left[\dfrac{3 + x}{3 - x}\right]^{\frac{1}{x}}
  8. Ex 7.6 II Q3
    Evaluate the following limit : lim⁡x→0[5x+33−2x]2x\lim_{x \to 0}\left[\dfrac{5x + 3}{3 - 2x}\right]^{\frac{2}{x}}
  9. Ex 7.6 II Q4
    Evaluate the following limit : lim⁡x→0[log⁡(3−x)−log⁡(3+x)x]\lim_{x \to 0}\left[\dfrac{\log(3 - x) - \log(3 + x)}{x}\right]
  10. Ex 7.6 II Q5
    Evaluate the following limit : lim⁡x→0[4x+11−4x]1x\lim_{x \to 0}\left[\dfrac{4x + 1}{1 - 4x}\right]^{\frac{1}{x}}
  11. Ex 7.6 II Q6
    Evaluate the following limit : lim⁡x→0[5+7x5−3x]13x\lim_{x \to 0}\left[\dfrac{5 + 7x}{5 - 3x}\right]^{\frac{1}{3x}}
  12. Ex 7.6 III Q1
    Evaluate the following limit : lim⁡x→0[ax−bxsin⁡(4x)−sin⁡(2x)]\lim_{x \to 0}\left[\dfrac{a^{x} - b^{x}}{\sin(4x) - \sin(2x)}\right]
  13. Ex 7.6 III Q2
    Evaluate the following limit : lim⁡x→0[(2x−1)3(3x−1)⋅sin⁡x⋅log⁡(1+x)]\lim_{x \to 0}\left[\dfrac{\left(2^{x} - 1\right)^{3}}{\left(3^{x} - 1\right) \cdot \sin x \cdot \log(1 + x)}\right]
  14. Ex 7.6 III Q3
    Evaluate the following limit : lim⁡x→0[15x−5x−3x+1x⋅sin⁡x]\lim_{x \to 0}\left[\dfrac{15^{x} - 5^{x} - 3^{x} + 1}{x \cdot \sin x}\right]
  15. Ex 7.6 III Q4
    Evaluate the following limit : lim⁡x→0[(25)x−2(5)x+1x⋅sin⁡x]\lim_{x \to 0}\left[\dfrac{(25)^{x} - 2(5)^{x} + 1}{x \cdot \sin x}\right]
  16. Ex 7.6 III Q5
    Evaluate the following limit : lim⁡x→0[(49)x−2(35)x+(25)xsin⁡x⋅log⁡(1+2x)]\lim_{x \to 0}\left[\dfrac{(49)^{x} - 2(35)^{x} + (25)^{x}}{\sin x \cdot \log(1 + 2x)}\right]

7.7 Limits at Infinity

3 q

Solved Examples

Worked · 3
  1. 7.7 SolvedEx.1
    Evaluate : lim⁡x→∞[ax+bcx+d]\lim_{x \to \infty}\left[\frac{ax+b}{cx+d}\right]
  2. 7.7 SolvedEx.2
    Evaluate : lim⁡x→∞[10x2+5x+35x2−3x+8]\lim_{x \to \infty}\left[\frac{10x^2+5x+3}{5x^2-3x+8}\right]
  3. 7.7 SolvedEx.3
    Evaluate : lim⁡x→∞[x2+3x−x]\lim_{x \to \infty}\left[\sqrt{x^2+3x} - x\right]

Exercise 7.7

11 q
  1. Ex 7.7 I Q1
    Evaluate the following : lim⁡x→∞[ax3+bx2+cx+dex3+fx2+gx+h]\lim_{x \to \infty}\left[\frac{ax^3 + bx^2 + cx + d}{ex^3 + fx^2 + gx + h}\right]
  2. Ex 7.7 I Q2
    Evaluate the following : lim⁡x→∞[x3+3x+2(x+4)(x−6)(x−3)]\lim_{x \to \infty}\left[\frac{x^3 + 3x + 2}{(x+4)(x-6)(x-3)}\right]
  3. Ex 7.7 I Q3
    Evaluate the following : lim⁡x→∞[7x2+5x−38x2−2x+7]\lim_{x \to \infty}\left[\frac{7x^2 + 5x - 3}{8x^2 - 2x + 7}\right]
  4. Ex 7.7 II Q1
    Evaluate the following : lim⁡x→∞[7x2+2x−3x4+x+2]\lim_{x \to \infty}\left[\frac{7x^2 + 2x - 3}{\sqrt{x^4 + x + 2}}\right]
  5. Ex 7.7 II Q2
    Evaluate the following : lim⁡x→∞[x2+4x+16−x2+16]\lim_{x \to \infty}\left[\sqrt{x^2 + 4x + 16} - \sqrt{x^2 + 16}\right]
  6. Ex 7.7 II Q3
    Evaluate the following : lim⁡x→∞[x4+4x2−x2]\lim_{x \to \infty}\left[\sqrt{x^4 + 4x^2} - x^2\right]
  7. Ex 7.7 III Q1
    Evaluate the following : lim⁡x→∞[(3x2+4)(4x2−6)(5x2+2)4x6+2x4−1]\lim_{x \to \infty}\left[\frac{(3x^2+4)(4x^2-6)(5x^2+2)}{4x^6 + 2x^4 - 1}\right]
  8. Ex 7.7 III Q2
    Evaluate the following : lim⁡x→∞[(3x−4)3(4x+3)4(3x+2)7]\lim_{x \to \infty}\left[\frac{(3x-4)^3 (4x+3)^4}{(3x+2)^7}\right]
  9. Ex 7.7 III Q3
    Evaluate the following : lim⁡x→∞[x(x+1−x)]\lim_{x \to \infty}\left[\sqrt{x}\left(\sqrt{x+1} - \sqrt{x}\right)\right]
  10. Ex 7.7 III Q4
    Evaluate the following : lim⁡x→∞[(2x−1)20(3x−1)30(2x+1)50]\lim_{x \to \infty}\left[\frac{(2x-1)^{20}(3x-1)^{30}}{(2x+1)^{50}}\right]
  11. Ex 7.7 III Q5
    Evaluate the following : lim⁡x→∞[x2+5−x2−3x2+3−x2+1]\lim_{x \to \infty}\left[\frac{\sqrt{x^2+5} - \sqrt{x^2-3}}{\sqrt{x^2+3} - \sqrt{x^2+1}}\right]

Miscellaneous Exercise 7

39 q

(I) Select the correct answer

Practice · 15
  1. Misc I Q1
    lim⁡x→2(x4−16x2−5x+6)=\lim_{x \to 2}\left(\frac{x^{4}-16}{x^{2}-5x+6}\right) =
    1. A.
      2323
    2. B.
      3232
    3. C.
      −32-32
    4. D.
      −16-16
  2. Misc I Q2
    lim⁡x→−2(x7+128x3+8)=\lim_{x \to -2}\left(\frac{x^{7}+128}{x^{3}+8}\right) =
    1. A.
      563\frac{56}{3}
    2. B.
      1123\frac{112}{3}
    3. C.
      1213\frac{121}{3}
    4. D.
      283\frac{28}{3}
  3. Misc I Q3
    lim⁡x→3(1x2−11x+24+1x2−x−6)=\lim_{x \to 3}\left(\frac{1}{x^{2}-11x+24}+\frac{1}{x^{2}-x-6}\right) =
    1. A.
      −225-\frac{2}{25}
    2. B.
      225\frac{2}{25}
    3. C.
      725\frac{7}{25}
    4. D.
      −725-\frac{7}{25}
  4. Misc I Q4
    lim⁡x→5(x+4−33x−11−2)=\lim_{x \to 5}\left(\frac{\sqrt{x+4}-3}{\sqrt{3x-11-2}}\right) =
    1. A.
      −29\frac{-2}{9}
    2. B.
      27\frac{2}{7}
    3. C.
      59\frac{5}{9}
    4. D.
      29\frac{2}{9}
  5. Misc I Q5
    lim⁡x→π3(tan⁡2x−3sec⁡3x−8)=\lim_{x \to \frac{\pi}{3}}\left(\frac{\tan^{2}x-3}{\sec^{3}x-8}\right) =
    1. A.
      11
    2. B.
      12\frac{1}{2}
    3. C.
      13\frac{1}{3}
    4. D.
      14\frac{1}{4}
  6. Misc I Q6
    lim⁡x→0(5sin⁡x−xcos⁡x2tan⁡x−3x2)=\lim_{x \to 0}\left(\frac{5\sin x - x\cos x}{2\tan x - 3x^{2}}\right) =
    1. A.
      00
    2. B.
      11
    3. C.
      22
    4. D.
      33
  7. Misc I Q7
    lim⁡x→π2[3cos⁡x+cos⁡3x(2x−π)3]=\lim_{x \to \frac{\pi}{2}}\left[\frac{3\cos x + \cos 3x}{(2x-\pi)^{3}}\right] =
    1. A.
      32\frac{3}{2}
    2. B.
      12\frac{1}{2}
    3. C.
      −12-\frac{1}{2}
    4. D.
      14\frac{1}{4}
  8. Misc I Q8
    lim⁡x→0(15x−3x−5x+1sin⁡2x)=\lim_{x \to 0}\left(\frac{15^{x}-3^{x}-5^{x}+1}{\sin^{2}x}\right) =
    1. A.
      log⁡15\log 15
    2. B.
      log⁡3+log⁡5\log 3 + \log 5
    3. C.
      log⁡3⋅log⁡5\log 3 \cdot \log 5
    4. D.
      3log⁡53\log 5
  9. Misc I Q9
    lim⁡x→0(3+5x3−4x)1x=\lim_{x \to 0}\left(\frac{3+5x}{3-4x}\right)^{\frac{1}{x}} =
    1. A.
      e3e^{3}
    2. B.
      e6e^{6}
    3. C.
      e9e^{9}
    4. D.
      e−3e^{-3}
  10. Misc I Q10
    lim⁡x→0[log⁡(5+x)−log⁡(5−x)sin⁡x]=\lim_{x \to 0}\left[\frac{\log(5+x)-\log(5-x)}{\sin x}\right] =
    1. A.
      32\frac{3}{2}
    2. B.
      −52-\frac{5}{2}
    3. C.
      −12-\frac{1}{2}
    4. D.
      25\frac{2}{5}
  11. Misc I Q11
    lim⁡x→π2(3cos⁡x−1π2−x)=\lim_{x \to \frac{\pi}{2}}\left(\frac{3^{\cos x}-1}{\frac{\pi}{2}-x}\right) =
    1. A.
      11
    2. B.
      log⁡3\log 3
    3. C.
      3π23^{\frac{\pi}{2}}
    4. D.
      3log⁡33\log 3
  12. Misc I Q12
    lim⁡x→0[x⋅log⁡(1+3x)(e3x−1)2]=\lim_{x \to 0}\left[\frac{x \cdot \log(1+3x)}{(e^{3x}-1)^{2}}\right] =
    1. A.
      1e9\frac{1}{e^{9}}
    2. B.
      1e3\frac{1}{e^{3}}
    3. C.
      19\frac{1}{9}
    4. D.
      13\frac{1}{3}
  13. Misc I Q13
    lim⁡x→0[(3sin⁡x−1)3(3x−1)⋅tan⁡x⋅log⁡(1+x)]=\lim_{x \to 0}\left[\frac{(3^{\sin x}-1)^{3}}{(3^{x}-1)\cdot \tan x \cdot \log(1+x)}\right] =
    1. A.
      3log⁡33\log 3
    2. B.
      2log⁡32\log 3
    3. C.
      (log⁡3)2(\log 3)^{2}
    4. D.
      (log⁡3)3(\log 3)^{3}
  14. Misc I Q14
    lim⁡x→3[5x−3−4x−3sin⁡(x−3)]=\lim_{x \to 3}\left[\frac{5^{x-3}-4^{x-3}}{\sin(x-3)}\right] =
    1. A.
      log⁡5−4\log 5 - 4
    2. B.
      log⁡54\log \frac{5}{4}
    3. C.
      log⁡5log⁡4\frac{\log 5}{\log 4}
    4. D.
      log⁡54\frac{\log 5}{4}
  15. Misc I Q15
    lim⁡x→∞[(2x+3)7(x−5)3(2x−5)10]=\lim_{x \to \infty}\left[\frac{(2x+3)^{7}(x-5)^{3}}{(2x-5)^{10}}\right] =
    1. A.
      38\frac{3}{8}
    2. B.
      18\frac{1}{8}
    3. C.
      16\frac{1}{6}
    4. D.
      14\frac{1}{4}

(II) Answer the following

Practice · 24
  1. Misc II Q1
    lim⁡x→0[(1−x)5−1(1−x)3−1]\lim_{x \to 0}\left[\frac{(1-x)^{5}-1}{(1-x)^{3}-1}\right]
  2. Misc II Q2
    lim⁡x→0[x]\lim_{x \to 0}[x] ([∗][\ast] is a greatest integer function.)
  3. Misc II Q3
    If f(r)=πr2f(r) = \pi r^{2} then find lim⁡h→0[f(r+h)−f(r)h]\lim_{h \to 0}\left[\frac{f(r+h)-f(r)}{h}\right]
  4. Misc II Q4
    lim⁡x→0[x∣x∣+x2]\lim_{x \to 0}\left[\frac{x}{|x|+x^{2}}\right]
  5. Misc II Q5
    Find the limit of the function, if it exists, at x=1x = 1 f(x)={7−4xfor x<1x2+2for x≥1f(x) = \begin{cases} 7-4x & \text{for } x<1 \\ x^{2}+2 & \text{for } x \ge 1 \end{cases}
  6. Misc II Q6
    Given that 7x≤f(x)≤3x2−67x \le f(x) \le 3x^{2}-6 for all xx. Determine the value of lim⁡x→3f(x)\lim_{x \to 3} f(x)
  7. Misc II Q7
    lim⁡x→0[sec⁡x2−1x4]\lim_{x \to 0}\left[\frac{\sec x^{2}-1}{x^{4}}\right]
  8. Misc II Q8
    lim⁡x→0[ex+e−x−2x⋅tan⁡x]\lim_{x \to 0}\left[\frac{e^{x}+e^{-x}-2}{x \cdot \tan x}\right]
  9. Misc II Q9
    lim⁡x→0[x(6x−3x)cos⁡(6x)−cos⁡(4x)]\lim_{x \to 0}\left[\frac{x(6^{x}-3^{x})}{\cos(6x)-\cos(4x)}\right]
  10. Misc II Q10
    lim⁡x→0[a3x−a2x−ax+1x⋅tan⁡x]\lim_{x \to 0}\left[\frac{a^{3x}-a^{2x}-a^{x}+1}{x \cdot \tan x}\right]
  11. Misc II Q11
    lim⁡x→a[sin⁡x−sin⁡ax−a]\lim_{x \to a}\left[\frac{\sin x - \sin a}{x-a}\right]
  12. Misc II Q12
    lim⁡x→2[log⁡x−log⁡2x−2]\lim_{x \to 2}\left[\frac{\log x - \log 2}{x-2}\right]
  13. Misc II Q13
    lim⁡x→1[abx−axbx2−1]\lim_{x \to 1}\left[\frac{ab^{x}-a^{x}b}{x^{2}-1}\right]
  14. Misc II Q14
    lim⁡x→0[(5x−1)2(2x−1)log⁡(1+x)]\lim_{x \to 0}\left[\frac{(5^{x}-1)^{2}}{(2^{x}-1)\log(1+x)}\right]
  15. Misc II Q15
    lim⁡x→∞[(2x+1)2(7x−3)3(5x+2)5]\lim_{x \to \infty}\left[\frac{(2x+1)^{2}(7x-3)^{3}}{(5x+2)^{5}}\right]
  16. Misc II Q16
    lim⁡x→a[xcos⁡a−acos⁡xx−a]\lim_{x \to a}\left[\frac{x\cos a - a\cos x}{x-a}\right]
  17. Misc II Q17
    lim⁡x→π4[(sin⁡x−cos⁡x)22−sin⁡x−cos⁡x]\lim_{x \to \frac{\pi}{4}}\left[\frac{(\sin x - \cos x)^{2}}{\sqrt{2}-\sin x - \cos x}\right]
  18. Misc II Q18
    lim⁡x→1[22x−2−2x+1sin⁡2(x−1)]\lim_{x \to 1}\left[\frac{2^{2x-2}-2^{x}+1}{\sin^{2}(x-1)}\right]
  19. Misc II Q19
    lim⁡x→1[4x−1−2x+1(x−1)2]\lim_{x \to 1}\left[\frac{4^{x-1}-2^{x}+1}{(x-1)^{2}}\right]
  20. Misc II Q20
    lim⁡x→1[x−1log⁡x]\lim_{x \to 1}\left[\frac{\sqrt{x}-1}{\log x}\right]
  21. Misc II Q21
    lim⁡x→0(1−cos⁡xx)\lim_{x \to 0}\left(\frac{\sqrt{1-\cos x}}{x}\right)
  22. Misc II Q22
    lim⁡x→1(x+3x2+5x3+⋯+(2n−1)xn−n2x−1)\lim_{x \to 1}\left(\frac{x+3x^{2}+5x^{3}+\cdots+(2n-1)x^{n}-n^{2}}{x-1}\right)
  23. Misc II Q23
    lim⁡x→0{1x12[1−cos⁡(x22)−cos⁡(x44)+cos⁡(x22)cos⁡(x44)]}\lim_{x \to 0}\left\{\frac{1}{x^{12}}\left[1-\cos\left(\frac{x^{2}}{2}\right)-\cos\left(\frac{x^{4}}{4}\right)+\cos\left(\frac{x^{2}}{2}\right)\cos\left(\frac{x^{4}}{4}\right)\right]\right\}
  24. Misc II Q24
    lim⁡x→∞(8x2+5x+32x2−7x−5)4x+38x−1\lim_{x \to \infty}\left(\frac{8x^{2}+5x+3}{2x^{2}-7x-5}\right)^{\frac{4x+3}{8x-1}}