Question

Finding limits: Find the limit, and use a graphing device to confirm your result graphically.

$\underset{x\to -1}{\lim }\frac{x^2-x-2}{x^3-x}$

Short answer

The limit of function is$-\frac{3}{2}$.

Step-by-step solution

Step 1. Write the given expression.

The expression is$\underset{x\to -1}{\lim }\frac{x^2-x-2}{x^3-x}$

Step 2. Use Concept of Indeterminate Forms.

Some forms of limits are called indeterminate if the limiting behavior of individual parts of the given expression is not able to determine the overall limit.

If we plug$x=-1$into given expression, we obtain$\frac{0}{0}$ , which is an Indeterminate form.

So, we need to do some calculations.

Step 3. Do the calculation.

$\begin{array}{c}\underset{x\to -1}{\lim }\frac{x^2-x-2}{x^3-x}=\underset{x\to -1}{\lim }\frac{x^2+x-2x-2}{x(x^2-1)}\\ =\underset{x\to -1}{\lim }\frac{x(x+1)-2(x+1)}{x(x-1)(x+1)}\\ =\underset{x\to -1}{\lim }\frac{(x-2)(x+1)}{x(x-1)(x+1)}\\ =\underset{x\to -1}{\lim }\frac{(x-2)}{x(x-1)}\end{array}$

Step 4. Apply the limit.

$\begin{array}{c}\underset{x\to -1}{\lim }\frac{x^2-x-2}{x^3-x}=\frac{(-1-2)}{-1(-1-1)}\\ =\frac{-3}{-1\times -2}\\ =-\frac{3}{2}\end{array}$

Step 5. Confirm the result using graph.

Use the graph calculate to plot the graph of function. So, we get

As we can see from the graph, as$x$approaches$-1$, graph of the function approaches to$-\frac{3}{2}=-\mathrm{1.5.}$

Therefore, the limit of function is $-\frac{3}{2}$.