To solve the limit [tex]\(\lim _{x \rightarrow \infty}(\sqrt{x-a}-\sqrt{x-b})\)[/tex] step-by-step, let's break down the expression and analyze it as [tex]\(x\)[/tex] approaches infinity.
We start with the expression:
[tex]\[
\lim_{x \to \infty} (\sqrt{x - a} - \sqrt{x - b})
\][/tex]
1. Rewrite the expression:
Consider the form [tex]\(\sqrt{x - a} - \sqrt{x - b}\)[/tex]. To facilitate the simplification, we multiply and divide by the conjugate, [tex]\(\sqrt{x - a} + \sqrt{x - b}\)[/tex]:
[tex]\[
\sqrt{x - a} - \sqrt{x - b} = \frac{(\sqrt{x - a} - \sqrt{x - b})(\sqrt{x - a} + \sqrt{x - b})}{\sqrt{x - a} + \sqrt{x - b}}
\][/tex]
2. Simplify the numerator:
Applying the difference of squares formula in the numerator, we get:
[tex]\[
(\sqrt{x - a} - \sqrt{x - b})(\sqrt{x - a} + \sqrt{x - b}) = (x - a) - (x - b)
\][/tex]
This simplifies to:
[tex]\[
(x - a) - (x - b) = -a + b = b - a
\][/tex]
3. Combine the terms:
Now we put the simplified numerator back over the denominator:
[tex]\[
\sqrt{x - a} - \sqrt{x - b} = \frac{b - a}{\sqrt{x - a} + \sqrt{x - b}}
\][/tex]
4. Consider the limit as [tex]\(x \to \infty\)[/tex]:
As [tex]\(x\)[/tex] approaches infinity, both [tex]\(\sqrt{x - a}\)[/tex] and [tex]\(\sqrt{x - b}\)[/tex] become very large because the terms [tex]\(a\)[/tex] and [tex]\(b\)[/tex] become negligible in comparison to [tex]\(x\)[/tex]. Therefore, [tex]\(\sqrt{x - a} \approx \sqrt{x}\)[/tex] and [tex]\(\sqrt{x - b} \approx \sqrt{x}\)[/tex].
Denominator can be approximated by:
[tex]\[
\sqrt{x - a} + \sqrt{x - b} \approx \sqrt{x} + \sqrt{x} = 2\sqrt{x}
\][/tex]
5. Substitute the approximation:
Replacing the denominator, we get:
[tex]\[
\frac{b - a}{\sqrt{x - a} + \sqrt{x - b}} \approx \frac{b - a}{2\sqrt{x}}
\][/tex]
6. Evaluate the limit:
Now, as [tex]\(x\)[/tex] approaches infinity, the term [tex]\(\frac{b - a}{2\sqrt{x}}\)[/tex] approaches zero because the denominator grows without bounds while the numerator remains constant:
[tex]\[
\lim_{x \to \infty} \frac{b - a}{2\sqrt{x}} = 0
\][/tex]
Thus, the final answer is:
[tex]\[
\lim_{x \to \infty} (\sqrt{x - a} - \sqrt{x - b}) = 0
\][/tex]
