Evaluate lim(x→0) (3^x - 3^ 3-x - 12)/(3^3-x - 3^x/2)

Question

Evaluate \(\lim\limits_{\text x \to2}\left(\cfrac{3^{\text x}-3^{3-\text x}-12}{3^{3-\text x}-3^{\text x/2}}\right) \)

lim(x→0) (3^x - 3^{3 - x} - 12)/(3^{3 - x} - 3^x/2)

Answer 1

To evaluate \(\lim\limits_{\text x \to2}\left(\cfrac{3^{\text x}-3^{3-\text x}-12}{3^{3-\text x}-3^{\text x/2}}\right) \)

lim(x→0) (3^x - 3^{3 - x} - 12)/(3^{3 - x} - 3^x/2)

Formula used: L'Hospital's rule Let f(x) and g(x) be two functions which are differentiable on an open interval I except at a point a where

This represents an indeterminate form. Thus applying L'Hospital's rule, we get

Thus, the value of \(\lim\limits_{\text x \to2}\left(\cfrac{3^{\text x}-3^{3-\text x}-12}{3^{3-\text x}-3^{\text x/2}}\right) \) is \(\cfrac{28\,In\,3}{-26.5\,In\,3}\)