QuestionFind the value of $k$ such that $\lim _{x \rightarrow 2} f(x)=3$ for $f(x)=\frac{(x-2)(x^{2}-k^{2})}{(x^{2}-4)(x-k)}$ .

Studdy Solution

STEP 1

Assumptions1. The function is defined as $f(x)=\frac{(x-)\left(x^{}-k^{}\right)}{\left(x^{}-4\right)(x-k)}$ . The constant is represented by $k$
3. We are looking for the value of $k$ such that $\lim{x \rightarrow} f(x)=3$

STEP 2

We know that the limit of a function as $x$ approaches a certain value is the value that the function approaches as $x$ gets closer and closer to that value. So, we need to find the value of $k$ that makes $f(x)$ approach $$ as $x$ approaches $2$.

STEP 3

First, we can simplify the function $f(x)$ by factoring out the terms in the numerator and the denominator.
$f(x)=\frac{(x-2)\left((x-k)(x+k)\right)}{(x-2)(x+2)(x-k)}$

STEP 4

We can cancel out the common factors $(x-2)$ and $(x-k)$ from the numerator and the denominator.
$f(x)=\frac{x+k}{x+2}$

STEP 5

Now, we substitute $x=2$ into the simplified function and set it equal to $3$ .
$3 = \frac{2+k}{2+2}$

STEP 6

olving the equation for $k$ gives us $3 = \frac{2+k}{4}$ $3 \times4 =2 + k$ $12 =2 + k$ $k =12 -2$ $k =10$ So, the value of $k$ that makes $\lim{x \rightarrow2} f(x)=3$ is $10$ .
The correct answer is (C)10.

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QuestionFind the value of $k$ such that $\lim _{x \rightarrow 2} f(x)=3$ for $f(x)=\frac{(x-2)(x^{2}-k^{2})}{(x^{2}-4)(x-k)}$ .

Studdy Solution

STEP 1

Assumptions1. The function is defined as $f(x)=\frac{(x-)\left(x^{}-k^{}\right)}{\left(x^{}-4\right)(x-k)}$ . The constant is represented by $k$
3. We are looking for the value of $k$ such that $\lim{x \rightarrow} f(x)=3$

STEP 2

We know that the limit of a function as $x$ approaches a certain value is the value that the function approaches as $x$ gets closer and closer to that value. So, we need to find the value of $k$ that makes $f(x)$ approach $$ as $x$ approaches $2$.

STEP 3

First, we can simplify the function $f(x)$ by factoring out the terms in the numerator and the denominator.
$f(x)=\frac{(x-2)\left((x-k)(x+k)\right)}{(x-2)(x+2)(x-k)}$

STEP 4

We can cancel out the common factors $(x-2)$ and $(x-k)$ from the numerator and the denominator.
$f(x)=\frac{x+k}{x+2}$

STEP 5

Now, we substitute $x=2$ into the simplified function and set it equal to $3$ .
$3 = \frac{2+k}{2+2}$

STEP 6

olving the equation for $k$ gives us $3 = \frac{2+k}{4}$ $3 \times4 =2 + k$ $12 =2 + k$ $k =12 -2$ $k =10$ So, the value of $k$ that makes $\lim{x \rightarrow2} f(x)=3$ is $10$ .
The correct answer is (C)10.

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QuestionFind the value of kkk such that lim⁡x→2f(x)=3\lim _{x \rightarrow 2} f(x)=3limx→2​f(x)=3 for f(x)=(x−2)(x2−k2)(x2−4)(x−k)f(x)=\frac{(x-2)(x^{2}-k^{2})}{(x^{2}-4)(x-k)}f(x)=(x2−4)(x−k)(x−2)(x2−k2)​.

Studdy Solution

STEP 1

STEP 2

We know that the limit of a function as xxx approaches a certain value is the value that the function approaches as xxx gets closer and closer to that value. So, we need to find the value of kkk that makes f(x)f(x)f(x) approach $$ as $x$ approaches $2$.

STEP 3

First, we can simplify the function f(x)f(x)f(x) by factoring out the terms in the numerator and the denominator.f(x)=(x−2)((x−k)(x+k))(x−2)(x+2)(x−k)f(x)=\frac{(x-2)\left((x-k)(x+k)\right)}{(x-2)(x+2)(x-k)}f(x)=(x−2)(x+2)(x−k)(x−2)((x−k)(x+k))​

STEP 4

We can cancel out the common factors (x−2)(x-2)(x−2) and (x−k)(x-k)(x−k) from the numerator and the denominator.f(x)=x+kx+2f(x)=\frac{x+k}{x+2}f(x)=x+2x+k​

STEP 5

Now, we substitute x=2x=2x=2 into the simplified function and set it equal to 333.3=2+k2+23 = \frac{2+k}{2+2}3=2+22+k​

STEP 6

olving the equation for kkk gives us3=2+k43 = \frac{2+k}{4}3=42+k​3×4=2+k3 \times4 =2 + k3×4=2+k12=2+k12 =2 + k12=2+kk=12−2k =12 -2k=12−2k=10k =10k=10So, the value of kkk that makes lim⁡x→2f(x)=3\lim{x \rightarrow2} f(x)=3limx→2f(x)=3 is 101010.The correct answer is (C)10.

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QuestionFind the value of $k$ such that $\lim _{x \rightarrow 2} f(x)=3$ for $f(x)=\frac{(x-2)(x^{2}-k^{2})}{(x^{2}-4)(x-k)}$ .

We know that the limit of a function as $x$ approaches a certain value is the value that the function approaches as $x$ gets closer and closer to that value. So, we need to find the value of $k$ that makes $f(x)$ approach $$ as $x$ approaches $2$.

First, we can simplify the function $f(x)$ by factoring out the terms in the numerator and the denominator.
$f(x)=\frac{(x-2)\left((x-k)(x+k)\right)}{(x-2)(x+2)(x-k)}$

We can cancel out the common factors $(x-2)$ and $(x-k)$ from the numerator and the denominator.
$f(x)=\frac{x+k}{x+2}$

Now, we substitute $x=2$ into the simplified function and set it equal to $3$ .
$3 = \frac{2+k}{2+2}$

olving the equation for $k$ gives us $3 = \frac{2+k}{4}$ $3 \times4 =2 + k$ $12 =2 + k$ $k =12 -2$ $k =10$ So, the value of $k$ that makes $\lim{x \rightarrow2} f(x)=3$ is $10$ .
The correct answer is (C)10.