QuestionCheck if the function $f(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{x-1} & \text{if } x \neq 1 \\ 4 & \text{if } x=1 \end{array}\right.$ is continuous at $a=1$ .

Studdy Solution

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.
$\lim_{x \rightarrow1} f(x) =1 +1 =2$

STEP 8

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 $\lim_{x \rightarrow1} f(x) \neq f(1)$ .

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QuestionCheck if the function $f(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{x-1} & \text{if } x \neq 1 \\ 4 & \text{if } x=1 \end{array}\right.$ is continuous at $a=1$ .

Studdy Solution

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.
$\lim_{x \rightarrow1} f(x) =1 +1 =2$

STEP 8

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QuestionCheck if the function f(x)={x2−1x−1if x≠14if x=1f(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{x-1} & \text{if } x \neq 1 \\ 4 & \text{if } x=1 \end{array}\right.f(x)={x−1x2−1​4​if x=1if x=1​ is continuous at a=1a=1a=1.

Studdy Solution

STEP 1

STEP 2

STEP 3

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

STEP 4

STEP 5

implify the function by cancelling out the common factor x−1x-1x−1.f(x)=x+1forx≠1f(x) = x +1 \quad \text{for} \quad x \neq1f(x)=x+1forx=1

STEP 6

Now, calculate the limit of f(x)f(x)f(x) as xxx approaches 111.lim⁡x→1f(x)=lim⁡x→1(x+1)\lim_{x \rightarrow1} f(x) = \lim_{x \rightarrow1} (x +1)x→1lim​f(x)=x→1lim​(x+1)

STEP 7

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

STEP 8

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QuestionCheck if the function $f(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{x-1} & \text{if } x \neq 1 \\ 4 & \text{if } x=1 \end{array}\right.$ is continuous at $a=1$ .

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

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

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

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