Question

3 - 13. determine the following limits at infinity.

1. $lim_{x

ightarrowinfty}x^{12}$

1. $lim_{x

ightarrow-infty}3x^{11}$

1. $lim_{x

ightarrowinfty}x^{-6}$

1. $lim_{x

ightarrow-infty}x^{-11}$

1. $lim_{t

ightarrowinfty}(- 12t^{-5})$

1. $lim_{x

ightarrow-infty}2x^{-8}$

1. $lim_{x

ightarrowinfty}(3+\frac{10}{x^{2}})$

1. $lim_{x

ightarrowinfty}(5+\frac{1}{x}+\frac{10}{x^{2}})$

Explanation:

Step1: Recall power - function limit rules

For $\lim_{x
ightarrow\infty}x^n$, if $n>0$, $\lim_{x
ightarrow\infty}x^n=\infty$; if $n < 0$, $\lim_{x
ightarrow\infty}x^n = 0$. For $\lim_{x
ightarrow-\infty}x^n$, if $n$ is even, $\lim_{x
ightarrow-\infty}x^n=\infty$, if $n$ is odd, $\lim_{x
ightarrow-\infty}x^n=-\infty$ when $n>0$, and $\lim_{x
ightarrow-\infty}x^n = 0$ when $n<0$. Also, $\lim_{x
ightarrow\pm\infty}c=c$ for a constant $c$.

Step2: Solve 3. $\lim_{x

ightarrow\infty}x^{12}$
Since $n = 12>0$, $\lim_{x
ightarrow\infty}x^{12}=\infty$.

Step3: Solve 4. $\lim_{x

ightarrow-\infty}3x^{11}$
Since $n = 11>0$ and odd, $\lim_{x
ightarrow-\infty}x^{11}=-\infty$, so $\lim_{x
ightarrow-\infty}3x^{11}=3\times(-\infty)=-\infty$.

Step4: Solve 5. $\lim_{x

ightarrow\infty}x^{-6}$
Since $n=-6 < 0$, $\lim_{x
ightarrow\infty}x^{-6}=\lim_{x
ightarrow\infty}\frac{1}{x^{6}} = 0$.

Step5: Solve 6. $\lim_{x

ightarrow-\infty}x^{-11}$
Since $n=-11 < 0$, $\lim_{x
ightarrow-\infty}x^{-11}=\lim_{x
ightarrow-\infty}\frac{1}{x^{11}} = 0$.

Step6: Solve 7. $\lim_{t

ightarrow\infty}(- 12t^{-5})$
Since $n=-5 < 0$, $\lim_{t
ightarrow\infty}(-12t^{-5})=-12\lim_{t
ightarrow\infty}\frac{1}{t^{5}} = 0$.

Step7: Solve 8. $\lim_{x

ightarrow-\infty}2x^{-8}$
Since $n=-8 < 0$, $\lim_{x
ightarrow-\infty}2x^{-8}=2\lim_{x
ightarrow-\infty}\frac{1}{x^{8}} = 0$.

Step8: Solve 9. $\lim_{x

ightarrow\infty}(3+\frac{10}{x^{2}})$
We know that $\lim_{x
ightarrow\infty}\frac{10}{x^{2}} = 0$ and $\lim_{x
ightarrow\infty}3 = 3$. By the sum - rule of limits $\lim_{x
ightarrow\infty}(3+\frac{10}{x^{2}})=\lim_{x
ightarrow\infty}3+\lim_{x
ightarrow\infty}\frac{10}{x^{2}}=3 + 0=3$.

Step9: Solve 10. $\lim_{x

ightarrow\infty}(5+\frac{1}{x}+\frac{10}{x^{2}})$
We know that $\lim_{x
ightarrow\infty}\frac{1}{x}=0$, $\lim_{x
ightarrow\infty}\frac{10}{x^{2}} = 0$ and $\lim_{x
ightarrow\infty}5 = 5$. By the sum - rule of limits $\lim_{x
ightarrow\infty}(5+\frac{1}{x}+\frac{10}{x^{2}})=\lim_{x
ightarrow\infty}5+\lim_{x
ightarrow\infty}\frac{1}{x}+\lim_{x
ightarrow\infty}\frac{10}{x^{2}}=5 + 0+0=5$.

Answer:

1. $\infty$
2. $-\infty$
3. $0$
4. $0$
5. $0$
6. $0$
7. $3$
8. $5$