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Section 2.8 : Limits at Infinity, Part II

9. Evaluate \(\displaystyle \mathop {\lim }\limits_{x \to \,\infty } \ln \left( {\frac{{11 + 8x}}{{{x^3} + 7x}}} \right)\).

Show Solution

First notice that,

\[\mathop {\lim }\limits_{x \to \,\infty } \frac{{11 + 8x}}{{{x^3} + 7x}} = 0\]

If you aren’t sure about this limit you should go back to the previous section and work some of the examples there to make sure that you can do these kinds of limits.

Also, note that because we are evaluating the limit \(x \to \infty \) it is safe to assume that \(x > 0\) and so we can further say that,

\[\frac{{11 + 8x}}{{{x^3} + 7x}} \to {0^ + }\]

Now, recalling Example 5 from this section, we know that because the argument goes to zero from the right in the limit the answer is,

\[\mathop {\lim }\limits_{x \to \,\infty } \ln \left( {\frac{{11 + 8x}}{{{x^3} + 7x}}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \infty }}\]