To find the limit [tex]\(\lim_{x \rightarrow \infty}(\sqrt{x-a}-\sqrt{x-b})\)[/tex], let's start by exploring the behavior of the function as [tex]\(x\)[/tex] approaches infinity.
1. Writing the expression in a more convenient form:
Begin with the original expression:
[tex]\[
\sqrt{x-a} - \sqrt{x-b}
\][/tex]
2. Rationalize the expression:
To simplify this, we'll multiply and divide by the conjugate of the expression (i.e., [tex]\(\sqrt{x-a} + \sqrt{x-b}\)[/tex]). This helps us remove the square roots from the numerator:
[tex]\[
\left(\sqrt{x-a} - \sqrt{x-b}\right) \times \frac{\sqrt{x-a} + \sqrt{x-b}}{\sqrt{x-a} + \sqrt{x-b}}
\][/tex]
Simplify the result:
[tex]\[
\frac{(\sqrt{x-a} - \sqrt{x-b})(\sqrt{x-a} + \sqrt{x-b})}{\sqrt{x-a} + \sqrt{x-b}}
\][/tex]
3. Simplify the numerator:
The numerator now becomes a difference of squares:
[tex]\[
(\sqrt{x-a} - \sqrt{x-b})(\sqrt{x-a} + \sqrt{x-b}) = (x-a) - (x-b) = -a + b = b-a
\][/tex]
So, we simplify our expression to:
[tex]\[
\frac{b-a}{\sqrt{x-a} + \sqrt{x-b}}
\][/tex]
4. Evaluate the denominator:
As [tex]\(x\)[/tex] approaches infinity, both [tex]\(\sqrt{x-a}\)[/tex] and [tex]\(\sqrt{x-b}\)[/tex] will behave similarly to [tex]\(\sqrt{x}\)[/tex], since [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants. Therefore:
[tex]\[
\sqrt{x-a} + \sqrt{x-b} \approx \sqrt{x} + \sqrt{x} = 2\sqrt{x}
\][/tex]
5. Substitute back and further simplify:
Plug the approximate values back into the expression:
[tex]\[
\frac{b-a}{2\sqrt{x}}
\][/tex]
6. Evaluate the limit:
Now, consider the limit as [tex]\(x\)[/tex] approaches infinity:
[tex]\[
\lim_{x \to \infty} \frac{b-a}{2\sqrt{x}}
\][/tex]
Since [tex]\(\sqrt{x}\)[/tex] increases without bound as [tex]\(x\)[/tex] goes to infinity, the fraction [tex]\(\frac{b-a}{2\sqrt{x}}\)[/tex] will approach 0. Thus, we have:
[tex]\[
\lim_{x \rightarrow \infty}(\sqrt{x-a}-\sqrt{x-b}) = 0
\][/tex]
So, the desired limit is:
[tex]\[
\boxed{0}
\][/tex]
