Answer
Continuous
Work Step by Step
We are given the function:
$f(x)=\dfrac{2x^2+3x+1}{x^2+5x}$
We use the continuity checklist to determine if $f$ is continuous in $a=5$:
1) $f(x)=\dfrac{2x^2+3x+1}{x^2+5x}=\dfrac{2x^2+2x+x+1}{x(x+5)}$
$=\dfrac{2x(x+1)+(x+1)}{x(x+5)}=\dfrac{(x+1)(2x+1)}{x(x+5)}$
$a=5$ is not a zero of the denominator, therefore $f$ is defined for $a=5$.
2) $\lim\limits_{x \to 5} f(x)=\lim\limits_{x \to 5} \dfrac{(x+1)(2x+1)}{x(x+5)}=\dfrac{(5+1)(2\cdot 5+1)}{5(5+5)}=\dfrac{66}{50}=\dfrac{33}{25}$
Therefore $\lim\limits_{x \to 5}$ exists.
3) $f(5)=\dfrac{(5+1)(2\cdot 5+1)}{5(5+5)}=\dfrac{66}{50}=\dfrac{33}{25}$
Therefore $\lim\limits_{x \to 5} f(x)=f(5)$
As the conditions 1, 2, 3 are satisfied, the function is continuous in $a=5$.
