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]
