Question

Find the limit of this expression please.

Answer 1

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.