Chapter 12, Problem 1CT

To determine

Find the limit algebraically.

Answer to Problem 1CT

The limit of the function is $\boxed{-0.75}$

Explanation of Solution

Given information:

Sketch a graph of the function and approximate the limit (if it exists). Then find the limit (if it exists) algebraically by using appropriate technique(s).

  $\underset{x\to -2}{\lim }\frac{x^2-1}{2x}$

Calculation:

Consider the given function,

  $\underset{x\to -2}{\lim }\frac{x^2-1}{2x}$

Sketch a graph of the function use graphing utility $TI-83$

Press Y and key and observe the screen,

  

Now press WINDOW to choose proper scale,

  

Now press GRAPH and display will be,

  

Now verify the existence of limit, Press 2^nd window and display will be.

  

Now enter the values of $X$ as $X=1.9,1.99,2,2.001,2.01,2.1$ and ENTER successively.

  

As the value of the function approaches to the same value as $X$ approaches from left and right the limit at $x=-2$ exists.

Now evaluate the limit of function by substitution $x=-2$ by direct substitute method.

  $\begin{array}{l}\underset{x\to -2}{\lim }\frac{x^2-1}{2x}\\ =\frac{{\left(-2\right)}^2-1}{2\left(-2\right)}\\ =\frac{4-1}{-4}=-\frac{3}{4}\\ =-0.75\end{array}$

Hence the limit of the function is $\boxed{-0.75}$