Sure, let's find the limit of the given expression as [tex]\( x \)[/tex] approaches infinity:
[tex]$ \lim_{{x \to \infty}} \frac{\sqrt{x} - 1 + \sqrt{x - 1}}{\sqrt{x^2 - 1}} $[/tex]
To tackle this limit, we can analyze the behavior of the numerator and the denominator as [tex]\( x \)[/tex] grows larger and see how they behave relative to each other.
First, let's simplify the components individually as [tex]\( x \to \infty \)[/tex]:
1. Expression in the Numerator:
- [tex]\(\sqrt{x} \)[/tex] behaves asymptotically like [tex]\(\sqrt{x} \)[/tex] because as [tex]\( x \)[/tex] approaches infinity, the dominant term is [tex]\(\sqrt{x} \)[/tex].
- [tex]\(\sqrt{x-1} \)[/tex] behaves similarly to [tex]\(\sqrt{x} \)[/tex] because [tex]\( x-1 \approx x \)[/tex] for very large [tex]\( x \)[/tex].
- Therefore, [tex]\( \sqrt{x} - 1 + \sqrt{x - 1} \approx \sqrt{x} + \sqrt{x} = 2\sqrt{x} \)[/tex].
2. Expression in the Denominator:
- Write [tex]\( \sqrt{x^2 - 1} \approx \sqrt{x^2} \)[/tex] because as [tex]\( x \)[/tex] becomes very large, [tex]\(-1\)[/tex] is negligible compared to [tex]\( x^2 \)[/tex].
- Thus, [tex]\( \sqrt{x^2 - 1} \approx \sqrt{x^2} = x \)[/tex].
Now we have simplified the expression:
[tex]\[ \frac{2\sqrt{x}}{x} \][/tex]
This can be simplified further:
[tex]\[ \frac{2\sqrt{x}}{x} = \frac{2}{\sqrt{x}} \][/tex]
Finally, let's analyze the simplified expression as [tex]\( x \)[/tex] approaches infinity:
[tex]\[ \lim_{{x \to \infty}} \frac{2}{\sqrt{x}} \][/tex]
As [tex]\( x \)[/tex] approaches infinity, [tex]\(\sqrt{x} \)[/tex] also approaches infinity. Hence,
[tex]\[ \frac{2}{\sqrt{x}} \][/tex] approaches 0.
So, the limit is:
[tex]\[ \lim_{{x \to \infty}} \frac{\sqrt{x} - 1 + \sqrt{x - 1}}{\sqrt{x^2 - 1}} = 0 \][/tex]
Therefore, the limit of the given expression as [tex]\( x \)[/tex] approaches infinity is:
[tex]\[ 0 \][/tex]
