Answer
Not continuous
Work Step by Step
We are given the function:
$f(x)=\begin{cases}
\dfrac{x^2+x}{x+1},\text{ if }x\not=-1\\
2,\text{ if }x=-1
\end{cases}$
We use the continuity checklist to determine if $f$ is continuous in $a=-1$:
Rewrite the function:
$f(x)=\begin{cases}
\dfrac{x(x+1)}{x+1},\text{ if }x\not=-1\\
2,\text{ if }x=-1
\end{cases}$
$f(x)=\begin{cases}
x,\text{ if }x\not=-1\\
2,\text{ if }x=-1
\end{cases}$
1) $f(x)$ is defined in $a=-1$.
2) $\lim\limits_{x \to -1^-} f(x)=\lim\limits_{x \to -1^-} x=-1$
$\lim\limits_{x \to -1^+} f(x)=\lim\limits_{x \to -1^+} x=-1$
Therefore $\lim\limits_{x \to -1}$ exists.
3) $f(-1)=2$
$\lim\limits_{x \to -1}=-1$
Therefore $\lim\limits_{x \to -1} f(x)\not=f(-1)$
As the condition 3 is not satisfied, the function is not continuous in $a=-1$.
