identify and describe the types of discontinuities in the following function: $f(x)=\begin{cases}x^{2}-4, &…

Explanation:

Step1: Find the limit as x approaches 2

We first find $\lim_{x\rightarrow2}(x^{2}-4)$. Using the direct - substitution for the non - piecewise part of the function for $x\neq2$. Since $y = x^{2}-4$ is a polynomial, $\lim_{x\rightarrow2}(x^{2}-4)=2^{2}-4=0$.

Step2: Evaluate the function at x = 2

The value of the function at $x = 2$ is given as $f(2)=3$.

Step3: Determine the type of discontinuity

Since $\lim_{x\rightarrow2}f(x)=0$ and $f(2) = 3$, and $\lim_{x\rightarrow2}f(x)$ exists but $\lim_{x\rightarrow2}f(x)\neq f(2)$, this is a removable discontinuity.

Answer:

The function $f(x)$ has a removable discontinuity at $x = 2$.