Math

QuestionEvaluate the limit: limx2x2+6x+4x2\lim _{x \rightarrow 2} \frac{x^{2}+6 x+4}{x-2}. If it doesn't exist, enter DNE.

Studdy Solution

STEP 1

Assumptions1. We are given the function f(x)=x+6x+4xf(x) = \frac{x^{}+6x+4}{x-}. We are asked to find the limit as x approaches.

STEP 2

First, try to directly substitute x=2 into the function.
f(2)=22+6(2)+422f(2) = \frac{2^{2}+6(2)+4}{2-2}

STEP 3

implify the numerator.
f(2)=+12+22f(2) = \frac{+12+}{2-2}

STEP 4

implify the denominator.
f(2)=200f(2) = \frac{20}{0}

STEP 5

We can't divide by zero, so the limit is undefined when directly substituting x=2. We need to try a different approach.

STEP 6

Rewrite the function by factoring the numerator.
f(x)=(x+2)(x+2)x2f(x) = \frac{(x+2)(x+2)}{x-2}

STEP 7

Notice that the numerator is a perfect square. Rewrite the function asf(x)=(x+2)2x2f(x) = \frac{(x+2)^2}{x-2}

STEP 8

Now, we can see that the function is undefined at x=2. This is because the denominator becomes zero at x=2, which makes the function undefined. However, we can still find the limit as x approaches2 by using L'Hopital's rule.

STEP 9

According to L'Hopital's rule, if the limit of a function is in the form of/ or ∞/∞, then the limit of that function is equal to the limit of the derivative of the numerator divided by the derivative of the denominator.

STEP 10

First, find the derivative of the numerator.
ddx(x+2)2=2(x+2)\frac{d}{dx} (x+2)^2 =2(x+2)

STEP 11

Next, find the derivative of the denominator.
ddx(x)=\frac{d}{dx} (x-) =

STEP 12

Now, substitute these derivatives back into the limit.
limx22(x+2)\lim{x \rightarrow2} \frac{2(x+2)}{}

STEP 13

implify the limit.
limx22(x+2)\lim{x \rightarrow2}2(x+2)

STEP 14

Now, substitute x=2 into the limit.
2(2+2)2(2+2)

STEP 15

implify the expression.
2(4)=82(4) =8So, the limit of the function as x approaches2 is8.

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