Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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]
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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.