Sure, let's find the limit of the expression [tex]\(\lim _{x \rightarrow \infty} \frac{1-\sqrt{x}}{1+\sqrt{x}}\)[/tex].
We need to determine the behavior of [tex]\(\frac{1-\sqrt{x}}{1+\sqrt{x}}\)[/tex] as [tex]\(x\)[/tex] approaches infinity.
1. Expression at the limit: When [tex]\(x\)[/tex] becomes very large, [tex]\(\sqrt{x}\)[/tex] becomes very large as well.
2. Simplification by factoring out [tex]\(\sqrt{x}\)[/tex]: We can simplify the given expression by dividing both the numerator and the denominator by [tex]\(\sqrt{x}\)[/tex], which will help us better understand the behavior as [tex]\(x\)[/tex] grows larger.
[tex]\[
\lim _{x \rightarrow \infty} \frac{1-\sqrt{x}}{1+\sqrt{x}}
\][/tex]
Dividing the numerator and the denominator by [tex]\(\sqrt{x}\)[/tex], we get:
[tex]\[
\frac{\frac{1}{\sqrt{x}} - 1}{\frac{1}{\sqrt{x}} + 1}
\][/tex]
3. Behavior of individual terms:
As [tex]\(x\)[/tex] approaches infinity:
- [tex]\(\frac{1}{\sqrt{x}}\)[/tex] approaches 0 because [tex]\(\sqrt{x}\)[/tex] becomes very large.
Substituting these limits into the simplified expression, we have:
[tex]\[
\frac{0 - 1}{0 + 1} = \frac{-1}{1} = -1
\][/tex]
Thus, the limit of the given expression as [tex]\(x\)[/tex] approaches infinity is:
[tex]\[
\boxed{-1}
\][/tex]
