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QuestionFind the limit: lim⁡x→22−xx+2−2\lim _{x \rightarrow 2} \frac{2-x}{\sqrt{x+2}-2} and simplify to lim⁡x→2(x+2+2x)\lim _{x \rightarrow 2}(\sqrt{x+2}+2 x).

Studdy Solution

STEP 1

Assumptions1. We are given the limit expression lim⁡x→−xx+−\lim{x \rightarrow} \frac{-x}{\sqrt{x+}-}. . We are asked to simplify the rational expression as much as possible.
3. We are given the limit expression lim⁡x→(x++x)\lim{x \rightarrow}(\sqrt{x+}+ x).

STEP 2

First, we need to simplify the first limit expression. We can do this by multiplying the numerator and denominator by the conjugate of the denominator.
lim⁡x→22−xx+2−2×x+2+2x+2+2\lim{x \rightarrow2} \frac{2-x}{\sqrt{x+2}-2} \times \frac{\sqrt{x+2}+2}{\sqrt{x+2}+2}

STEP 3

The multiplication of the denominator by its conjugate simplifies to a difference of squares.
lim⁡x→2(2−x)(x+2+2)(x+2)−\lim{x \rightarrow2} \frac{(2-x)(\sqrt{x+2}+2)}{(x+2)-}

STEP 4

implify the denominator to getlim⁡x→2(2−x)(x+2+2)x−2\lim{x \rightarrow2} \frac{(2-x)(\sqrt{x+2}+2)}{x-2}

STEP 5

Now, cancel out the common factors in the numerator and the denominator.
lim⁡x→2−(x+2+2)\lim{x \rightarrow2} -(\sqrt{x+2}+2)

STEP 6

Now, we substitute x=2x =2 into the simplified limit expression.
−2+2−2=−22−2-\sqrt{2+2} -2 = -2\sqrt{2} -2

STEP 7

Now, we simplify the second limit expression. Substitute x=2x =2 into the expression.
lim⁡x→2(x+2+2x)=2+2+2×2=22+4\lim{x \rightarrow2}(\sqrt{x+2}+2 x) = \sqrt{2+2} +2 \times2 =2\sqrt{2} +4So, the simplified expressions for the given limit problems are −22−2-2\sqrt{2} -2 and 22+42\sqrt{2} +4 respectively.

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