evaluate the limit using lhospitals rule\nlim(x→0) (13^x - 5^x)/x
evaluate the limit using lhospitals rule\nlim(x→0) (13^x - 5^x)/x
Answer
Explanation:
Step1: Check indeterminate form
When (x = 0), (\frac{13^{x}-5^{x}}{x}=\frac{13^{0}-5^{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 = 13^{x}-5^{x}) using the formula ((a^{x})^\prime=a^{x}\ln a) is (y^\prime=13^{x}\ln13 - 5^{x}\ln5), and the derivative of (y = x) is (y^\prime = 1).
Step3: Evaluate new limit
(\lim_{x\rightarrow0}\frac{13^{x}\ln13 - 5^{x}\ln5}{1}). Substitute (x = 0) into the expression: (13^{0}\ln13-5^{0}\ln5=\ln13-\ln5).
Answer:
(\ln13-\ln5)