Chapter 14.2, Problem 25AYU

To determine

To find: The value of the given limit.

Answer to Problem 25AYU

The value of the limit is $-1$ .

Explanation of Solution

Given:

The given limit is $\underset{x\to 0}{\lim }\frac{x^2-4}{x^2+4}$

Calculation:

Consider the given limit is,

  $\underset{x\to 0}{\lim }\frac{x^2-4}{x^2+4}$

Consider the function as,

  $\underset{x\to 0}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to 0}{\lim }f\left(x\right)}{\underset{x\to 0}{\lim }g\left(x\right)}$

Then,

  $\begin{array}{c}\underset{x\to 0}{\lim }\frac{x^2-4}{x^2+4}=\frac{\underset{x\to 0}{\lim }\left(x^2-4\right)}{\underset{x\to 0}{\lim }\left(x^2+4\right)}\\ =\frac{\underset{x\to 0}{\lim }\left({\left(0\right)}^2-4\right)}{\underset{x\to 0}{\lim }\left({\left(0\right)}^2+4\right)}\\ =\frac{-4}{4}\\ =-1\end{array}$