To solve the limit [tex]\(\lim_{x \rightarrow \infty}(\sqrt{x-a}-\sqrt{x-b})\)[/tex], we will walk through a series of algebraic manipulations and simplifications.
1. Start with the original limit expression:
[tex]\[
\lim_{x \rightarrow \infty} (\sqrt{x-a} - \sqrt{x-b})
\][/tex]
2. Rationalize the expression:
To handle the difference of square roots, we multiply and divide the expression by its conjugate pair:
[tex]\[
\lim_{x \rightarrow \infty} \frac{(\sqrt{x-a} - \sqrt{x-b})(\sqrt{x-a} + \sqrt{x-b})}{\sqrt{x-a} + \sqrt{x-b}}
\][/tex]
3. Simplify the numerator:
Notice that the numerator ([tex]\(\sqrt{x-a} - \sqrt{x-b})(\sqrt{x-a} + \sqrt{x-b})\)[/tex]) is a difference of squares:
[tex]\[
(\sqrt{x-a})^2 - (\sqrt{x-b})^2 = (x-a) - (x-b)
\][/tex]
This simplifies to:
[tex]\[
(x-a) - (x-b) = b - a
\][/tex]
4. Simplify the expression:
Now, we have:
[tex]\[
\lim_{x \rightarrow \infty} \frac{b-a}{\sqrt{x-a} + \sqrt{x-b}}
\][/tex]
5. Approximate the square roots for large [tex]\(x\)[/tex]:
As [tex]\(x\)[/tex] approaches infinity, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] become negligible in comparison to [tex]\(x\)[/tex]. Thus, we can approximate:
[tex]\[
\sqrt{x-a} \approx \sqrt{x} \quad \text{and} \quad \sqrt{x-b} \approx \sqrt{x}
\][/tex]
6. Substitute the approximations back:
[tex]\[
\lim_{x \rightarrow \infty} \frac{b-a}{\sqrt{x} + \sqrt{x}} = \lim_{x \rightarrow \infty} \frac{b-a}{2\sqrt{x}}
\][/tex]
7. Evaluate the limit:
Since [tex]\(\sqrt{x}\)[/tex] grows unbounded as [tex]\(x\)[/tex] approaches infinity, the fraction [tex]\(\frac{1}{\sqrt{x}}\)[/tex] approaches 0. Therefore:
[tex]\[
\lim_{x \rightarrow \infty} \frac{b-a}{2\sqrt{x}} = 0
\][/tex]
Thus, the evaluated limit is:
[tex]\[
\boxed{0}
\][/tex]
