Determine whether f is continuous at c.
-$$f ( x ) = \left\{ \begin{array} { c l }
\frac { 1 } { x - 2 } , & x 2 \
x ^ { 2 } + 5 x , & x \leq 2
\end{array} ; \quad c = 2 \right.$$
A) not continuous
B) continuous

Question

Determine whether f is continuous at c.
-$$f ( x ) = \left\{ \begin{array} { c l } 
\frac { 1 } { x - 2 } , & x > 2 \
x ^ { 2 } + 5 x , & x \leq 2
\end{array} ; \quad c = 2 \right.$$
A) not continuous
B) continuous

Quizplus · Accepted Answer

For a function to be continuous at a point $c$, the limit as $x$ approaches $c$ from both sides must exist and be equal to the function's value at $c$. Here, as $x$ approaches 2 from the left ($x < 2$), the function approaches $2^2 + 5(2) = 4 + 10 = 14$. As $x$ approaches 2 from the right ($x > 2$), the function $\frac{1}{x-2}$ approaches infinity, indicating a discontinuity. Since the limits from both sides do not match, $f$ is not continuous at $c = 2$.

Determine Whether F Is Continuous at C f(x)={1x−2,x>2x2+5x,x≤2;c=2f ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x - 2 } , & x > 2 \\x ^ { 2 } + 5 x , & x \leq 2\end{array} ; \quad c = 2 \right.f(x)={x−21​,x2+5x,​x>2x≤2​;c=2

Determine Whether F Is Continuous at C $f ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x - 2 } , & x > 2 \\x ^ { 2 } + 5 x , & x \leq 2\end{array} ; \quad c = 2 \right.$