Question

Provide all detailed steps to find the limit of the following functions.

Lim x→∞ (e^4x - e^-2x) ÷ (ln(x+1)

Answer 1

$$\lim\limits_{x\to \infin}\dfrac{e^{4x}-e^{-2x}}{\ln(x+1)}$$

Use L'Hôpital's rule

$$\lim\limits_{x\to \infin}\dfrac{e^{4x}-e^{-2x}}{\ln(x+1)}=\lim\limits_{x\to \infin}\dfrac{(e^{4x}-e^{-2x})'}{(\ln(x+1))'}$$

$$=\lim\limits_{x\to \infin}\dfrac{4e^{4x}+2e^{-2x}}{1/(x+1)}=\lim\limits_{x\to \infin}(4e^{4x}+2e^{-2x})(x+1)$$

$$=\infin$$