QuestionFind the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as $x$ approaches3 of the function $\frac{x^{}-x-6}{x-3}$ . . We are dealing with real numbers.

STEP 2

First, we try to substitute $x =$ into the function to see if we can directly compute the limit.
$\lim{x \rightarrow} \frac{x^{2}-x-6}{x-} = \frac{^{2}--6}{-}$

STEP 3

Calculate the value after substitution.
$\frac{3^{2}-3-6}{3-3} = \frac{9-3-6}{0}$

STEP 4

implify the numerator.
$\frac{9-3-6}{0} = \frac{0}{0}$

STEP 5

We see that direct substitution leads to an indeterminate form ( $\frac{0}{0}$ ). This means we need to do some algebraic manipulation to the function to find the limit. In this case, we can factor the numerator.

STEP 6

Factor the numerator $x^{2}-x-6$ .
$x^{2}-x-6 = (x-3)(x+2)$

STEP 7

Substitute the factored form of the numerator back into the function.
$\lim{x \rightarrow3} \frac{(x-3)(x+2)}{x-3}$

STEP 8

We see that the term $(x-3)$ appears in both the numerator and the denominator, so we can cancel it out.
$\lim{x \rightarrow3} x+2$

STEP 9

Now that we have simplified the function, we can substitute $x =3$ into the function to find the limit.
$\lim{x \rightarrow3} x+2 =3+2$

STEP 10

Calculate the value after substitution.
$3+2 =5$ So, the limit of the function as $x$ approaches3 is5.

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QuestionFind the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as $x$ approaches3 of the function $\frac{x^{}-x-6}{x-3}$ . . We are dealing with real numbers.

STEP 2

First, we try to substitute $x =$ into the function to see if we can directly compute the limit.
$\lim{x \rightarrow} \frac{x^{2}-x-6}{x-} = \frac{^{2}--6}{-}$

STEP 3

Calculate the value after substitution.
$\frac{3^{2}-3-6}{3-3} = \frac{9-3-6}{0}$

STEP 4

implify the numerator.
$\frac{9-3-6}{0} = \frac{0}{0}$

STEP 5

We see that direct substitution leads to an indeterminate form ( $\frac{0}{0}$ ). This means we need to do some algebraic manipulation to the function to find the limit. In this case, we can factor the numerator.

STEP 6

Factor the numerator $x^{2}-x-6$ .
$x^{2}-x-6 = (x-3)(x+2)$

STEP 7

Substitute the factored form of the numerator back into the function.
$\lim{x \rightarrow3} \frac{(x-3)(x+2)}{x-3}$

STEP 8

We see that the term $(x-3)$ appears in both the numerator and the denominator, so we can cancel it out.
$\lim{x \rightarrow3} x+2$

STEP 9

Now that we have simplified the function, we can substitute $x =3$ into the function to find the limit.
$\lim{x \rightarrow3} x+2 =3+2$

STEP 10

Calculate the value after substitution.
$3+2 =5$ So, the limit of the function as $x$ approaches3 is5.

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QuestionFind the limit: lim⁡x→3x2−x−6x−3\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}limx→3​x−3x2−x−6​.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as xxx approaches3 of the function x−x−6x−3\frac{x^{}-x-6}{x-3}x−3x−x−6​. . We are dealing with real numbers.

STEP 2

First, we try to substitute x=x =x= into the function to see if we can directly compute the limit.lim⁡x→x2−x−6x−=2−−6−\lim{x \rightarrow} \frac{x^{2}-x-6}{x-} = \frac{^{2}--6}{-}limx→x−x2−x−6​=−2−−6​

STEP 3

Calculate the value after substitution.32−3−63−3=9−3−60\frac{3^{2}-3-6}{3-3} = \frac{9-3-6}{0}3−332−3−6​=09−3−6​

STEP 4

implify the numerator.9−3−60=00\frac{9-3-6}{0} = \frac{0}{0}09−3−6​=00​

STEP 5

We see that direct substitution leads to an indeterminate form (00\frac{0}{0}00​). This means we need to do some algebraic manipulation to the function to find the limit. In this case, we can factor the numerator.

STEP 6

Factor the numerator x2−x−6x^{2}-x-6x2−x−6.x2−x−6=(x−3)(x+2)x^{2}-x-6 = (x-3)(x+2)x2−x−6=(x−3)(x+2)

STEP 7

Substitute the factored form of the numerator back into the function.lim⁡x→3(x−3)(x+2)x−3\lim{x \rightarrow3} \frac{(x-3)(x+2)}{x-3}limx→3x−3(x−3)(x+2)​

STEP 8

We see that the term (x−3)(x-3)(x−3) appears in both the numerator and the denominator, so we can cancel it out.lim⁡x→3x+2\lim{x \rightarrow3} x+2limx→3x+2

STEP 9

Now that we have simplified the function, we can substitute x=3x =3x=3 into the function to find the limit.lim⁡x→3x+2=3+2\lim{x \rightarrow3} x+2 =3+2limx→3x+2=3+2

STEP 10

Calculate the value after substitution.3+2=53+2 =53+2=5So, the limit of the function as xxx approaches3 is5.

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QuestionFind the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$ .

Assumptions1. We are asked to find the limit as $x$ approaches3 of the function $\frac{x^{}-x-6}{x-3}$ . . We are dealing with real numbers.

First, we try to substitute $x =$ into the function to see if we can directly compute the limit.
$\lim{x \rightarrow} \frac{x^{2}-x-6}{x-} = \frac{^{2}--6}{-}$

Calculate the value after substitution.
$\frac{3^{2}-3-6}{3-3} = \frac{9-3-6}{0}$

implify the numerator.
$\frac{9-3-6}{0} = \frac{0}{0}$

We see that direct substitution leads to an indeterminate form ( $\frac{0}{0}$ ). This means we need to do some algebraic manipulation to the function to find the limit. In this case, we can factor the numerator.

Factor the numerator $x^{2}-x-6$ .
$x^{2}-x-6 = (x-3)(x+2)$

Substitute the factored form of the numerator back into the function.
$\lim{x \rightarrow3} \frac{(x-3)(x+2)}{x-3}$

We see that the term $(x-3)$ appears in both the numerator and the denominator, so we can cancel it out.
$\lim{x \rightarrow3} x+2$

Now that we have simplified the function, we can substitute $x =3$ into the function to find the limit.
$\lim{x \rightarrow3} x+2 =3+2$

Calculate the value after substitution.
$3+2 =5$ So, the limit of the function as $x$ approaches3 is5.