evaluate the limit using lhospitals rule lim(x→0) (5^x - 11^x)/x

Explanation:

Step1: Check indeterminate form

As $x\to0$, we have $\frac{5^{0}-11^{0}}{0}=\frac{1 - 1}{0}=\frac{0}{0}$, so L'Hopital's rule can be applied.

Step2: Differentiate numerator and denominator

The derivative of $y = 5^{x}-11^{x}$ using the formula $(a^{x})^\prime=a^{x}\ln a$ is $y^\prime=5^{x}\ln 5-11^{x}\ln 11$, and the derivative of $y = x$ is $y^\prime = 1$.

Step3: Evaluate new - limit

$\lim_{x\to0}\frac{5^{x}\ln 5-11^{x}\ln 11}{1}$. Substitute $x = 0$ into the expression: $5^{0}\ln 5-11^{0}\ln 11=\ln 5-\ln 11$.

Answer:

$\ln 5-\ln 11$