Question

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 5^{+}} \ln \frac{x}{\sqrt{x-4}} $$

Short answer

The limit of the given function as \(x\) approaches 5 from the positive side is \( \ln 5 \).

Step-by-step solution

(Changing the log property)

The function given can be rewritten using the logarithm property, \( \ln m/n = \ln m - \ln n \). This gives \( \ln x - \ln \sqrt{x - 4} \).

(Simplify the terms)

Note that \( \ln \sqrt{x - 4} \) can be rewritten as \( 0.5 \ln(x - 4) \). So the function further simplifies to \( \ln x - 0.5 \ln(x - 4) \). This will make it easier to compute the limit.

(Evaluate the limit)

Next, resolve the limit \( \lim_{x \rightarrow 5^{+}} \ln x - 0.5 \lim_{x \rightarrow 5^{+}} \ln(x - 4) \). Since \( \lim_{x \rightarrow a} \ln b(x) \) is equal to \( \ln b(a) \), we find that \( \lim_{x \rightarrow 5^{+}} \ln x - 0.5 \lim_{x \rightarrow 5^{+}} \ln(x - 4) = \ln 5 - 0.5 \ln 1 \).

(Final result)

Logarithm of 1 is 0. So, the final result is \( \ln 5 - 0 \), which is simply \( \ln 5 \).