To determine the limit [tex]\(\lim_{{x \to 2^{+}}} \frac{x-2}{\sqrt{x-2}}\)[/tex], we need to carefully analyze the behavior of the function as [tex]\(x\)[/tex] approaches 2 from the right.
Here is a step-by-step solution:
1. Rewrite the Expression:
The given limit is:
[tex]\[
\lim_{{x \to 2^{+}}} \frac{x-2}{\sqrt{x-2}}
\][/tex]
Observe that both the numerator [tex]\(x - 2\)[/tex] and the denominator [tex]\(\sqrt{x-2}\)[/tex] involve the term [tex]\(x - 2\)[/tex].
2. Simplify the Expression:
Factorize the numerator in terms of the denominator:
[tex]\[
\frac{x-2}{\sqrt{x-2}} = \frac{y}{\sqrt{y}} \quad \text{where} \quad y = x - 2
\][/tex]
As [tex]\(x \to 2^{+}\)[/tex], [tex]\(y\)[/tex] approaches 0 from the right (i.e., [tex]\(y \to 0^{+}\)[/tex]).
3. Simplify Further:
The expression [tex]\(\frac{y}{\sqrt{y}}\)[/tex] can be simplified as follows:
[tex]\[
\frac{y}{\sqrt{y}} = \frac{y}{y^{1/2}} = y^{1 - 1/2} = y^{1/2} = \sqrt{y}
\][/tex]
4. Take the Limit:
Now, we take the limit of [tex]\(\sqrt{y}\)[/tex] as [tex]\(y \to 0^{+}\)[/tex]:
[tex]\[
\lim_{{y \to 0^{+}}} \sqrt{y}
\][/tex]
The square root function [tex]\(\sqrt{y}\)[/tex] approaches 0 as [tex]\(y\)[/tex] approaches 0 from the right.
5. Conclusion:
Therefore, the limit is:
[tex]\[
\lim_{{x \to 2^{+}}} \frac{x-2}{\sqrt{x-2}} = \lim_{{y \to 0^{+}}} \sqrt{y} = 0
\][/tex]
So, we conclude that:
[tex]\[
\lim_{{x \to 2^{+}}} \frac{x-2}{\sqrt{x-2}} = 0
\][/tex]
