Answered: "Find the limits:limx→−2x+2x2+5−3"

ICESSRAIPCELOtblt

Answered question

2022-03-29

Find the limits:

lim_{x \to - 2} \frac{x + 2}{\sqrt{x^{2} + 5} - 3}

Payten Reese

Beginner2022-03-30Added 10 answers

Given limits:

lim_{x \to - 2} \frac{x + 2}{\sqrt{x^{2} + 5} - 3}

Rationalize the denominator, we get

lim_{x \to - 2} \frac{x + 2}{\sqrt{x^{2} + 5} - 3} \times \frac{\sqrt{x^{2} + 5} + 3}{\sqrt{x^{2} + 5} + 3}

lim_{x \to - 2} \frac{(x + 2) (\sqrt{x^{2} + 5} + 3)}{{(\sqrt{x^{2} + 5})}^{2} - 3^{2}}

lim_{x \to - 2} \frac{(x + 2) (\sqrt{x^{2} + 5} + 3)}{x^{2} + 5 - 9}

lim_{x \to - 2} \frac{(x + 2) (\sqrt{x^{2} + 5} + 3)}{x^{2} - 4}

lim_{x \to - 2} \frac{(x + 2) (\sqrt{x^{2} + 5} + 3)}{x^{2} - 2^{2}}

lim_{x \to - 2} \frac{(x + 2) (\sqrt{x^{2} + 5} + 3)}{(x - 2) (x + 2)}

lim_{x \to - 2} \frac{(\sqrt{x^{2} + 5} + 3)}{(x - 2)}

Apply the limits x=-2, we get

lim_{x \to - 2} \frac{(\sqrt{x^{2} + 5} + 3)}{(x - 2)} = \frac{(\sqrt{{(- 2)}^{2} + 5} + 3} {(- 2 - 2)}}{}

lim_{x \to - 2} \frac{(\sqrt{x^{2} + 5} + 3)}{(x - 2)} = \frac{(\sqrt{4 + 5} + 3} {(- 2 - 2)}}{}

lim_{x \to - 2} \frac{(\sqrt{x^{2} + 5} + 3)}{(x - 2)} = \frac{(3 + 3)}{(- 2 - 2)}

lim_{x \to - 2} \frac{(\sqrt{x^{2} + 5} + 3)}{(x - 2)} = - \frac{3}{2}

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