Chapter 9.2, Problem 1E

To determine

To Find: The value of a limit, using both graphical method and L'Hôpital's Rule.

Answer to Problem 1E

The value of the limit is $\frac{1}{4}$ .

Explanation of Solution

Given Information: The limit is $\underset{x\to 2}{\lim }\frac{x-2}{x^2-4}$

Calculation:

The two method is which is to be used to find the value of the given limit is explained below:

1. Graphically: Plot the graph of the expression in $x$ and identify the value to which the expression approaches as $x$ approaches to $2$ .
2. $\underset{\_}{\text{L'H<mover accent="true"><mo> o </mo><mo>^</mo></mover> pital's Rule}}\text{: If }f(a)=g(a)=0,\text{ then }\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$

Therefore,

Using graphing utility, the graph of the function is obtained as:

  

In the above graph, the value of the expression approaches to $\frac{1}{4}$ near $x=2$ .

Therefore, the value of the limit is $\frac{1}{4}$ .

Now,

Apply L'Hôpital's Rule to write

  $\underset{x\to 2}{\lim }\left(\frac{x-2}{x^2-4}\right)=\underset{x\to 2}{\lim }\left(\frac{1-0}{2x-0}\right)=\underset{x\to 2}{\lim }\left(\frac{1}{2x}\right)=\frac{1}{2\times 2}=\frac{1}{4}$

The limit determined using both the methods is indeed the same.