Chapter 12, Problem 26RE

To determine

Apply a graphing utility to verify the result graphically.

Answer to Problem 26RE

Limit of the function $\underset{x\to 5}{\lim }\frac{5-x}{x^2-25}$ is $\boxed{\frac{-1}{10}}$

Explanation of Solution

Given information:

Find the limit (if it exists). Use a graphing utility to verify the result graphically.

  $\underset{x\to 5}{\lim }\frac{5-x}{x^2-25}$

Calculation:

The limit of given function is,

  $\underset{x\to 5}{\lim }\frac{5-x}{x^2-25}$

To determine the limit of a function follow these techniques.

1. Apply direct substitution method if the function is a polynomial.
2. Apply direct substitution method if the function is a rational function not in the form of $0/0$ .
3. Apply dividing out method if the function is a rational function in the form of $0/0$ .
4. Apply direct substitution method if the function is a irrational function not in the form of $0/0$ .
5. Apply direct substitution method if the function is a irrational function in the form of $0/0$ .

By applying technique $3$ we get,

  $\underset{x\to 5}{\lim }\frac{5-x}{x^2-25}$

Use algebraic identity on denominator of the given function,

  $\left(a^2-b^2\right)=\left(a+b\right)\left(a-b\right)$

  $\begin{array}{l}=\underset{x\to 5}{\lim }\frac{5-x}{x^2-25}=\underset{x\to 5}{\lim }\frac{-\left(x-5\right)}{\left(x-5\right)\left(x+5\right)}\\ =\frac{-1}{\left(x+5\right)}\end{array}$

Substitute $x=5$

  $\begin{array}{l}=\frac{-1}{\left(5+5\right)}\\ =\frac{-1}{10}\end{array}$

To verify the solution graphically use graphic utility Ti-83.

Press Y key in the function, the display will be,

  

Now press window key and choose the proper scale, the display will be.

  

Now press graph key, the display will be.

  

Now press second graph key, the display will be.

  

Hence limit of the function $\underset{x\to 5}{\lim }\frac{5-x}{x^2-25}$ is $\boxed{\frac{-1}{10}}$