Chapter 7.1, Problem 66E

To determine

To determine the given function is continuous or discontinuous.

Answer to Problem 66E

Given function is continuous at $x=5$

Explanation of Solution

Given information :

  $2x^2+7x-4=0$

Every polynomial is continuous on real line.

Let us prove the given using limit definition of continuity which says a function $f\left(x\right)$ is continuous on point $a$ if $f\left(a\right)$ and $\underset{x\to a}{\lim }f\left(x\right)$ are equal.

Now we find $f\left(5\right)$

  $f\left(5\right)=3\left(125\right)+2\left(5\right)-5=380$

  $\begin{array}{c}f\left(5+h\right)=3{\left(5+h\right)}^3+2\left(5+h\right)-5\\ =\left(5+h\right)\left(3{\left(5+h\right)}^2+2\right)-5\\ =\left(5+h\right)\left(77+3h^2+30h\right)-5\\ =380+45h^2+227h+3h^3\end{array}$

Now $380+45h^2+227h+3h^3\to 380\text{ as }h\to 0$

Hence $\underset{h\to 0}{\lim }\left(f\left(5+h\right)\right)=f\left(5\right)$

Hence it is continuous.