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(8^x - 4^x - 2^x + 1)/x^2 for x > 0 joined to e^x sin x + x + lambda log 4: find e^lambda

The same question, asked 2 times: 21 Apr 2025 S1*, 21 Apr 2025 S2. Every wording below, oldest first.

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  1. Question 1

    21 Apr 2025 S1

    143
    If f(x)={8x−4x−2x+1x2, if x>0exsin⁡x+x+λlog⁡4, if x⩽0f(x) =\left\{ \begin{matrix} \frac{8^{x}-4^{x}-2^{x}+ 1}{x^{2}}, & \text{~if~}x> 0 \\ e^{x}\sin x+x+\lambda \log4, & \text{~if~}x\leqslant 0 \end{matrix} \right. is continuous at x=0x= 0 then the value of 1000eλ=1000e^{\lambda}=

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  2. Question 2

    21 Apr 2025 S2

    109
    If f(x)={8x−4x−2x+1xx2, if x>0exsin⁡x+ix+λlog⁡4, if x⩽0,i∈Rf(x) =\left\{ \begin{matrix} \frac{8^{x}-4^{x}-2^{x}+1^{x}}{x^{2}}, & \text{~if~}x> 0 \\ e^{x}\sin x+ix+\lambda \log4, & \text{~if~}x\leqslant 0,i\in\mathbb{R} \end{matrix} \right. is continuous at x=0x= 0, then the value of 500eλ500e^{\lambda} is

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