Question

use the given graphs of the function f (left, in blue) and g (right, in red) to find the following limits:

1. $lim_{x

ightarrow1}f(x)+g(x) = 2$

1. $lim_{x

ightarrow2}f(x)+g(x) = 4$

1. $lim_{x

ightarrow0}f(x)g(x)=4$

1. $lim_{x

ightarrow0}\frac{f(x)}{g(x)} = 0$

1. $lim_{x

ightarrow1}sqrt{3 + f(x)}=sqrt{3}$

Explanation:

Step1: Recall limit - sum rule

$\lim_{x
ightarrow a}[f(x)+g(x)]=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)$

Step2: Recall limit - product rule

$\lim_{x
ightarrow a}[f(x)g(x)]=\lim_{x
ightarrow a}f(x)\cdot\lim_{x
ightarrow a}g(x)$

Step3: Recall limit - quotient rule

$\lim_{x
ightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x
ightarrow a}f(x)}{\lim_{x
ightarrow a}g(x)}$ (when $\lim_{x
ightarrow a}g(x)
eq0$)

Step4: Recall limit - composition rule for square - root

$\lim_{x
ightarrow a}\sqrt{h(x)}=\sqrt{\lim_{x
ightarrow a}h(x)}$ (when $\lim_{x
ightarrow a}h(x)\geq0$)

Step5: Estimate limits from graphs

For $\lim_{x
ightarrow1}[f(x) + g(x)]$, find $\lim_{x
ightarrow1}f(x)$ and $\lim_{x
ightarrow1}g(x)$ from the graphs and add them.
For $\lim_{x
ightarrow2}[f(x)+g(x)]$, find $\lim_{x
ightarrow2}f(x)$ and $\lim_{x
ightarrow2}g(x)$ from the graphs and add them.
For $\lim_{x
ightarrow0}[f(x)g(x)]$, find $\lim_{x
ightarrow0}f(x)$ and $\lim_{x
ightarrow0}g(x)$ from the graphs and multiply them.
For $\lim_{x
ightarrow0}\frac{f(x)}{g(x)}$, find $\lim_{x
ightarrow0}f(x)$ and $\lim_{x
ightarrow0}g(x)$ from the graphs and divide them.
For $\lim_{x
ightarrow1}\sqrt{3 + f(x)}$, first find $\lim_{x
ightarrow1}f(x)$, then add 3 and take the square - root.

Answer:

1. $\lim_{x

ightarrow1}[f(x)+g(x)] = 2$

1. $\lim_{x

ightarrow2}[f(x)+g(x)] = 4$

1. $\lim_{x

ightarrow0}[f(x)g(x)] = 4$

1. $\lim_{x

ightarrow0}\frac{f(x)}{g(x)} = 0$

1. $\lim_{x

ightarrow1}\sqrt{3 + f(x)}=\sqrt{3}$