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QuestionFind the limit as xx approaches 2 for the expression x2+x−6x2−4\frac{x^{2}+x-6}{x^{2}-4}.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit of the function x+x−6x−4\frac{x^{}+x-6}{x^{}-4} as xx approaches. . We will use the algebraic method to find the limit.

STEP 2

First, we will try to simplify the function. We can do this by factoring the numerator and the denominator.
The numerator x2+x−6x^{2}+x-6 can be factored into (x−2)(x+)(x-2)(x+).
The denominator x2−4x^{2}-4 can be factored into (x−2)(x+2)(x-2)(x+2).
So, the function can be written asx2+x−6x2−4=(x−2)(x+)(x−2)(x+2)\frac{x^{2}+x-6}{x^{2}-4} = \frac{(x-2)(x+)}{(x-2)(x+2)}

STEP 3

Now, we can cancel out the common factor (x−2)(x-2) in the numerator and the denominator.
(x−2)(x+3)(x−2)(x+2)=x+3x+2\frac{(x-2)(x+3)}{(x-2)(x+2)} = \frac{x+3}{x+2}

STEP 4

Now, we can find the limit of the simplified function as xx approaches2.
lim⁡x→2x+3x+2\lim{x \rightarrow2} \frac{x+3}{x+2}

STEP 5

Substitute x=2x =2 into the simplified function.
lim⁡x→2x+3x+2=2+32+2\lim{x \rightarrow2} \frac{x+3}{x+2} = \frac{2+3}{2+2}

STEP 6

Calculate the limit.
lim⁡x→2x+3x+2=54=1.25\lim{x \rightarrow2} \frac{x+3}{x+2} = \frac{5}{4} =1.25So, the limit of the function x2+x−6x2−4\frac{x^{2}+x-6}{x^{2}-4} as xx approaches2 is1.25.

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