Math Snap

STEP 1

Assumptions1. The function is defined as$$f(x)=\left\{\begin{array}{ll} \frac{x^{}-1}{x-1} & \text { if } x \neq1 \\ 4 & \text { if } x=1\end{array}\right.
$$. We are asked to determine if the function is continuous at $a=1$.

STEP 2

A function is continuous at a point $a$ if the following three conditions are met1. $f(a)$ is defined.
2. $\lim_{x \rightarrow a} f(x)$ exists.
. $\lim_{x \rightarrow a} f(x) = f(a)$.
We will check each of these conditions for the function $f(x)$ at $a=1$.

STEP 3

First, check if $f(a)$ is defined. According to the function definition, $f(1) =$.

STEP 4

Next, check if $\lim_{x \rightarrow a} f(x)$ exists. To do this, we need to simplify the function for $x \neq1$.
The function for $x \neq1$ is $\frac{x^{2}-1}{x-1}$. This can be factored as $\frac{(x-1)(x+1)}{x-1}$.

STEP 5

implify the function by cancelling out the common factor $x-1$.
$$f(x) = x +1 \quad \text{for} \quad x \neq1$$

STEP 6

Now, calculate the limit of $f(x)$ as $x$ approaches $1$.
$$\lim_{x \rightarrow1} f(x) = \lim_{x \rightarrow1} (x +1)$$

STEP 7

Substitute $x =1$ into the simplified function to find the limit.
$$\lim_{x \rightarrow1} f(x) =1 +1 =2$$

Finally, check if $\lim_{x \rightarrow a} f(x) = f(a)$.We have $\lim_{x \rightarrow1} f(x) =2$ and $f(1) =4$. Since these two values are not equal, the function is not continuous at $a=1$.
The correct answer is D. The function is not continuous at $a=1$ because $\lim_{x \rightarrow1} f(x) \neq f(1)$.