Question

Determine the infinite limit.

\(\mathop {\lim }\limits_{x \to {5^ - }} \frac{{x + 1}}{{x - 5}}\)

Short answer

The infinite limit is \(\mathop {\lim }\limits_{x \to {5^ - }} \frac{{x + 1}}{{x - 5}} = - \infty \).

Step-by-step solution

Intuitive Definition of an infinite Limit

Consider \(f\) as the function defined on both sides of \(a\), except possibly at \(a\) itself. Then \(\mathop {\lim }\limits_{x \to a} f\left( x \right) = \infty \) .

This means that the values of \(f\left( x \right)\) arbitrarily large by restricting \(x\) sufficiently close to \(a\) but not equal to\(a\).

Definition for the one-sided infinite limits as shown below:

\(\begin{aligned}\mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) &= \infty \,\,\,\,\,\,\,\,\,\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) &=& \infty \\\mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) &= - \infty \,\,\,\,\,\,\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) &=& - \infty \end{aligned}\)

Determine the infinite limit

The infinite limit is \(\mathop {\lim }\limits_{x \to {5^ - }} \frac{{x + 1}}{{x - 5}} = - \infty \) because the numerator of the function is positive, and the denominator approaches 0 from the negativeside \(x \to {5^ - }\).

Thus, the infinite limit is \(\mathop {\lim }\limits_{x \to {5^ - }} \frac{{x + 1}}{{x - 5}} = - \infty \).