Thank you for visiting Guess the value of the limit if it exists by evaluating the function at the given numbers correct to 5 decimal places tex y 36. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
To solve this problem, we're tasked with finding the value of the limit:
[tex]\[
\lim _{y \rightarrow 36} \frac{36-y}{6-\sqrt{y}}
\][/tex]
We'll start by substituting different values of [tex]\( y \)[/tex] that are close to 36, both from above and below, to see how the function behaves.
### Approaching [tex]\( y \)[/tex] from above:
1. When [tex]\( y = 36.1 \)[/tex]:
[tex]\[
\frac{36 - 36.1}{6 - \sqrt{36.1}} \approx 12.00833
\][/tex]
2. When [tex]\( y = 36.01 \)[/tex]:
[tex]\[
\frac{36 - 36.01}{6 - \sqrt{36.01}} \approx 12.00083
\][/tex]
3. When [tex]\( y = 36.001 \)[/tex]:
[tex]\[
\frac{36 - 36.001}{6 - \sqrt{36.001}} \approx 12.00008
\][/tex]
4. When [tex]\( y = 36.0001 \)[/tex]:
[tex]\[
\frac{36 - 36.0001}{6 - \sqrt{36.0001}} \approx 12.00001
\][/tex]
### Approaching [tex]\( y \)[/tex] from below:
1. When [tex]\( y = 35.9 \)[/tex]:
[tex]\[
\frac{36 - 35.9}{6 - \sqrt{35.9}} \approx 11.99166
\][/tex]
2. When [tex]\( y = 35.99 \)[/tex]:
[tex]\[
\frac{36 - 35.99}{6 - \sqrt{35.99}} \approx 11.99917
\][/tex]
3. When [tex]\( y = 35.999 \)[/tex]:
[tex]\[
\frac{36 - 35.999}{6 - \sqrt{35.999}} \approx 11.99992
\][/tex]
4. When [tex]\( y = 35.9999 \)[/tex]:
[tex]\[
\frac{36 - 35.9999}{6 - \sqrt{35.9999}} \approx 11.99999
\][/tex]
### Conclusion:
As [tex]\( y \)[/tex] approaches 36 from both sides (above and below), the values of the fractional expression get closer to 12. From the calculations, it's evident that:
- When [tex]\( y \)[/tex] approaches 36 from above, the function value approaches 12.
- When [tex]\( y \)[/tex] approaches 36 from below, the function value also approaches 12.
Therefore, we can conclude that the limit exists and is equal to [tex]\( 12 \)[/tex].
So, the limit is:
[tex]\[
\lim _{y \rightarrow 36} \frac{36-y}{6-\sqrt{y}} = 12
\][/tex]
[tex]\[
\lim _{y \rightarrow 36} \frac{36-y}{6-\sqrt{y}}
\][/tex]
We'll start by substituting different values of [tex]\( y \)[/tex] that are close to 36, both from above and below, to see how the function behaves.
### Approaching [tex]\( y \)[/tex] from above:
1. When [tex]\( y = 36.1 \)[/tex]:
[tex]\[
\frac{36 - 36.1}{6 - \sqrt{36.1}} \approx 12.00833
\][/tex]
2. When [tex]\( y = 36.01 \)[/tex]:
[tex]\[
\frac{36 - 36.01}{6 - \sqrt{36.01}} \approx 12.00083
\][/tex]
3. When [tex]\( y = 36.001 \)[/tex]:
[tex]\[
\frac{36 - 36.001}{6 - \sqrt{36.001}} \approx 12.00008
\][/tex]
4. When [tex]\( y = 36.0001 \)[/tex]:
[tex]\[
\frac{36 - 36.0001}{6 - \sqrt{36.0001}} \approx 12.00001
\][/tex]
### Approaching [tex]\( y \)[/tex] from below:
1. When [tex]\( y = 35.9 \)[/tex]:
[tex]\[
\frac{36 - 35.9}{6 - \sqrt{35.9}} \approx 11.99166
\][/tex]
2. When [tex]\( y = 35.99 \)[/tex]:
[tex]\[
\frac{36 - 35.99}{6 - \sqrt{35.99}} \approx 11.99917
\][/tex]
3. When [tex]\( y = 35.999 \)[/tex]:
[tex]\[
\frac{36 - 35.999}{6 - \sqrt{35.999}} \approx 11.99992
\][/tex]
4. When [tex]\( y = 35.9999 \)[/tex]:
[tex]\[
\frac{36 - 35.9999}{6 - \sqrt{35.9999}} \approx 11.99999
\][/tex]
### Conclusion:
As [tex]\( y \)[/tex] approaches 36 from both sides (above and below), the values of the fractional expression get closer to 12. From the calculations, it's evident that:
- When [tex]\( y \)[/tex] approaches 36 from above, the function value approaches 12.
- When [tex]\( y \)[/tex] approaches 36 from below, the function value also approaches 12.
Therefore, we can conclude that the limit exists and is equal to [tex]\( 12 \)[/tex].
So, the limit is:
[tex]\[
\lim _{y \rightarrow 36} \frac{36-y}{6-\sqrt{y}} = 12
\][/tex]
Thank you for reading the article Guess the value of the limit if it exists by evaluating the function at the given numbers correct to 5 decimal places tex y 36. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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