Question

What value is a discontinuity of x^2+5x+2/x^2+2x-35

Answer 1

Solution:

Given the expression:

$\frac{x^2+5x+2}{2x^2+2x-35}$

A function f(x) has disconituity at x=a if

$\lim_{x\to a}f(x)$

exists and is finite.

The function is thus undefined at x=a or when

$\lim_{x\to a}(f(x))\ne f(a)$

From the given function, we have

$\begin{gathered} \frac{x^2+5x+2}{x^2+2x-35} \\ factorize\text{ the denominator,} \\ \frac{x^2+5x+2}{x^2-5x+7x-35}=\frac{x^2+5x+2}{x(x-5)+7(x-5)} \\ \Rightarrow\frac{x^2+5x+2}{(x-5)(x+7)} \end{gathered}$

The function is undefined at

$\begin{gathered} x-5=0 \\ \Rightarrow x=5 \\ x+7=0 \\ \Rightarrow x=-7 \end{gathered}$

Hence, there is discontinuity at

$x=5,\text{ x=-7}$