Let $f ( x ) = \left\{ \begin{array} { c c }
- x ^ { 2 } + 2 & - 3

Question

Let $f ( x ) = \left\{ \begin{array} { c c } 
- x ^ { 2 } + 2 & - 3 < x < 0 \
3 & x = 0 \
2 x + 4 & 0 < x \leq 3
\end{array} \right.$ Which of the following is true for $\lim _ { x \rightarrow 0 } f ( x )$ ?

A)2
B)3
C)4
D)0
E)Does not exist

Quizplus · Accepted Answer

The limit of $f(x)$ as $x$ approaches 0 from the left is $f(-0) = -0^2 + 2 = 2$, and the limit as $x$ approaches 0 from the right is $f(0+) = 2(0) + 4 = 4$. Since these two one-sided limits are not equal, the overall limit of $f(x)$ as $x$ approaches 0 does not exist.

Let f(x)={−x2+2−3<x<03x=02x+40<x≤3f ( x ) = \left\{ \begin{array} { c c } - x ^ { 2 } + 2 & - 3 < x < 0 \\3 & x = 0 \\2 x + 4 & 0 < x \leq 3\end{array} \right.f(x)=⎩⎨⎧​−x2+232x+4​−3<x<0x=00<x≤3​

Let $f ( x ) = \left\{ \begin{array} { c c } - x ^ { 2 } + 2 & - 3 < x < 0 \\3 & x = 0 \\2 x + 4 & 0 < x \leq 3\end{array} \right.$