Question

In Exercises 69-71, find the limit. Give a convincing argument that the value is correct. $$\lim _{x \rightarrow \infty} \frac{\ln x}{\log x}$$

Short answer

The limit as x approaches infinity of \( \frac{\ln x}{\log x} \) is \( \ln 10 \).

Step-by-step solution

Identifying the form and using L’Hospital’s Rule

The given function is \( \frac{\ln x}{\log x} \) as x tends towards infinity. This is an indeterminate form of the type ∞/∞. Hence, apply L’Hospital’s Rule which states that \(\frac{f(x)}{g(x)} \) as x goes to a value is equal to \( \frac{f'(x)}{g'(x)} \). Apply the rule to the given function. Also remember, \( \frac{d}{dx} (\ln x) = \frac{1}{x} \) and \( \frac{d}{dx} (\log_{10} x) = \frac{1}{x \ln 10} \).

Evaluate the derivatives' limit

After differentiating numerator and denominator, the limit becomes \lim_{x \rightarrow \infty} \frac{1/x}{1/(x \ln 10)} = \lim_{x \rightarrow \infty} \frac{\ln 10}{1}. Since \(\ln 10\) is a constant, as \(x\) tends to \(\infty\), the constant remains the same.

Final evaluation

The limit of a constant as \(x\) approaches any number (including \(\infty\)) is simply that constant. Hence, \( \lim_{x \rightarrow \infty} \frac{\ln x}{\log x} = \ln 10.\)