Question

use a graph or a table to find the following limit.\\(\lim\limits_{x \to \infty} \log_{7}(x)\\)\\(\lim\limits_{x \to \infty} \log_{7}(x) = \square\\) (simplify your answer.)

Explanation:

Step1: Recall Logarithm Properties

For a logarithmic function \( y = \log_b(x) \) where \( b>1 \), as \( x \) approaches \( \infty \), the function's value also approaches \( \infty \). Here, \( b = 7>1 \), so \( \log_7(x) \) is an increasing function.

Step2: Analyze Limit as \( x \to \infty \)

As \( x \) gets larger and larger (approaches \( \infty \)), the value of \( \log_7(x) \) will keep increasing without bound. So, \( \lim_{x \to \infty} \log_7(x)=\infty \).

Answer:

\(\infty\)