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Find the limit of this expression please.

Find the limit of this expression please.-example-1
User Kair
by
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1 Answer

3 votes

Answer: -4

Step-by-step explanation

If you plug in x = 2, then what results is the indeterminate form 0/0.

To avoid this error, we rationalize the denominator.

Multiply top and bottom by
√(x+2)+2

Here are the steps


(2-x)/(√(x+2)-2)\\\\=((2-x)(√(x+2)+2))/((√(x+2)-2)((√(x+2)+2)))\\\\=((2-x)(√(x+2)+2))/((√(x+2))^2-2^2) \ \ \text{ ..... difference of squares rule}\\\\=((2-x)(√(x+2)+2))/((x+2)-4)\\\\=((2-x)(√(x+2)+2))/(x-2)\\\\=-((x-2)(√(x+2)+2))/(x-2)\\\\=-(√(x+2)+2)\\\\=-√(x+2)-2\\\\

In short, we have the identity


(2-x)/(√(x+2)-2)=-√(x+2)-2, \text{ when } x \ge -2 \text{ and } x \\e 2\\\\

Put another way, the expression
(2-x)/(√(x+2)-2) simplifies to
-√(x+2)-2

We can now evaluate the limit.


\displaystyle L = \lim_(x \to 2)(2-x)/(√(x+2)-2)\\\\\displaystyle L = \lim_(x \to 2)-√(x+2)-2\\\\\displaystyle L = -√(2+2)-2\\\\\displaystyle L = -4\\\\

Therefore,


\displaystyle \lim_(x \to 2)(2-x)/(√(x+2)-2)=-4\\\\

This can be confirmed with a graph. GeoGebra and Desmos are two options I recommend. Other tools like WolframAlpha can be used to confirm the answer.

User RSW
by
8.3k points

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