Evaluate the limit using L'Hospital's rule \[ \lim _{x \rightarrow \infty} 13 x e^{\frac{1}{x}}-13 x \]
Solution
Step 1 ：Given the limit expression \(\lim _{x \rightarrow \infty} 13 x e^{\frac{1}{x}}-13 x\), we notice that it is in the indeterminate form of type \(\infty - \infty\).
Step 2 ：To apply L'Hospital's rule, we need to rewrite the expression in a form where we can apply the rule. We can do this by factoring out \(13x\) from the expression, which gives us \(13x(e^{\frac{1}{x}} - 1)\).
Step 3 ：Now, as \(x \rightarrow \infty\), the expression becomes an indeterminate form of type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), which is suitable for applying L'Hospital's rule.
Step 4 ：L'Hospital's rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives. So, we need to find the derivative of the numerator and the denominator and then find the limit.
Step 5 ：The numerator of our expression is \(e^{\frac{1}{x}} - 1\) and the denominator is \(\frac{1}{x}\).
Step 6 ：The derivative of the numerator is \(-\frac{e^{\frac{1}{x}}}{x^2}\) and the derivative of the denominator is \(-\frac{1}{x^2}\).
Step 7 ：Applying L'Hospital's rule, we find that the limit of the expression as \(x \rightarrow \infty\) is 1.
Step 8 ：Final Answer: \(\boxed{1}\)
