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Chapter 4: Problem 87

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty}(\sqrt{x-2}-\sqrt{x-4})$$

Short Answer

Expert verified
Answer: The limit of the function as $x$ tends to infinity is 0.

Step by step solution

01

Simplify the expression

To simplify the given expression, let's multiply and divide by the conjugate of the numerator: $$\lim_{x\rightarrow\infty}{\frac{(\sqrt{x-2}-\sqrt{x-4})(\sqrt{x-2}+\sqrt{x-4})}{(\sqrt{x-2}+\sqrt{x-4})}}$$
02

Apply difference of squares

Now we have difference of squares in the numerator, so we can evaluate it as follows: $$\lim_{x\rightarrow\infty}{\frac{(x-2)-(x-4)}{(\sqrt{x-2}+\sqrt{x-4})}}$$
03

Simplify the numerator

Simplify the numerator by combining like terms: $$\lim_{x\rightarrow\infty}{\frac{2}{(\sqrt{x-2}+\sqrt{x-4})}}$$
04

Substitute infinity and evaluate the limit

As x tends to infinity, both square root terms in the denominator will also tend to infinity. Therefore, the expression will have the form: $$\frac{2}{\infty+\infty}$$ This simplifies to 0. So, the limit is: $$\lim_{x\rightarrow\infty}{\frac{2}{(\sqrt{x-2}+\sqrt{x-4})}} = 0$$

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