Chapter 12, Problem 18RE

To determine

Evaluating a limit by direct substitution.

Answer to Problem 18RE

Limit of the function $\underset{t\to -2}{\lim }\left(5-2x-x^2\right)$ is $\boxed{5}$

Explanation of Solution

Given information:

Find the limit by direct substitution.

  $\underset{t\to -2}{\lim }\left(5-2x-x^2\right)$

Calculation:

The limit of given function is, $\underset{t\to -2}{\lim }\left(5-2x-x^2\right)$

Replace the $x$ by defined value.

By using property of limit,

  $\begin{array}{l}\underset{x\to c}{\lim }\left[f\left(x\right)\pm g\left(x\right)\right]\\ =\underset{x\to c}{\lim }f\left(x\right)\pm \underset{x\to c}{\lim }g\left(x\right)\end{array}$

  $\underset{x\to c}{\lim }kf\left(x\right)=k\underset{x\to c}{\lim }f\left(x\right)$

  $\underset{x\to c}{\lim }b=b$

Therefore,

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

Substitute $x=-2$

  $\begin{array}{l}\underset{x\to -2}{\lim }\left(5-2x-x^2\right)=5-2\left(-2\right)-{\left(-2\right)}^2\\ =5+4-4\\ =5\end{array}$

Hence limit of the function $\underset{t\to -2}{\lim }\left(5-2x-x^2\right)$ is $\boxed{5}$