use the given graphs of the function f (in blue) and g (in red) to find the following limits:\n1. $lim_{x…
use the given graphs of the function f (in blue) and g (in red) to find the following limits:\n1. $lim_{x \to 1}f(x)+g(x)=$\n2. $lim_{x \to 2}f(x)+g(x)=$\n3. $lim_{x \to 0}f(x)g(x)=$\n4. $lim_{x \to 0}\frac{f(x)}{g(x)}=$\n5. $lim_{x \to - 1}sqrt{3 + f(x)}=$\nnote: type inf for $infty$ and -inf for $-infty$. if the limit does not exist, write dne.\nyou can click on the graphs to enlarge the images.\nnote: in order to get credit for this problem all answers must be correct.
Answer
Explanation:
Step1: Recall limit - sum rule
$\lim_{x\rightarrow a}(f(x)+g(x))=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$; $\lim_{x\rightarrow a}(f(x)g(x))=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow a}g(x)$; $\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a}g(x)}$ (when $\lim_{x\rightarrow a}g(x)\neq0$); $\lim_{x\rightarrow a}\sqrt{h(x)}=\sqrt{\lim_{x\rightarrow a}h(x)}$ (when $\lim_{x\rightarrow a}h(x)\geq0$).
Step2: Find $\lim_{x\rightarrow 1}(f(x)+g(x))$
First, find $\lim_{x\rightarrow 1}f(x)$ and $\lim_{x\rightarrow 1}g(x)$ from the graphs. $\lim_{x\rightarrow 1}f(x) = 0$ and $\lim_{x\rightarrow 1}g(x)=0$. Then $\lim_{x\rightarrow 1}(f(x)+g(x))=\lim_{x\rightarrow 1}f(x)+\lim_{x\rightarrow 1}g(x)=0 + 0=0$.
Step3: Find $\lim_{x\rightarrow 2}(f(x)+g(x))$
Find $\lim_{x\rightarrow 2}f(x)$ and $\lim_{x\rightarrow 2}g(x)$ from the graphs. $\lim_{x\rightarrow 2}f(x)=0$ and $\lim_{x\rightarrow 2}g(x)=1$. Then $\lim_{x\rightarrow 2}(f(x)+g(x))=\lim_{x\rightarrow 2}f(x)+\lim_{x\rightarrow 2}g(x)=0 + 1=1$.
Step4: Find $\lim_{x\rightarrow 0}(f(x)g(x))$
Find $\lim_{x\rightarrow 0}f(x)$ and $\lim_{x\rightarrow 0}g(x)$ from the graphs. $\lim_{x\rightarrow 0}f(x)= - 1$ and $\lim_{x\rightarrow 0}g(x)=0$. Then $\lim_{x\rightarrow 0}(f(x)g(x))=\lim_{x\rightarrow 0}f(x)\cdot\lim_{x\rightarrow 0}g(x)=(-1)\times0 = 0$.
Step5: Find $\lim_{x\rightarrow 0}\frac{f(x)}{g(x)}$
Find $\lim_{x\rightarrow 0}f(x)= - 1$ and $\lim_{x\rightarrow 0}g(x)=0$. Since the denominator's limit is 0 and the numerator's limit is non - zero, $\lim_{x\rightarrow 0}\frac{f(x)}{g(x)}=\text{DNE}$.
Step6: Find $\lim_{x\rightarrow - 1}\sqrt{3 + f(x)}$
Find $\lim_{x\rightarrow - 1}f(x)$ from the graph. $\lim_{x\rightarrow - 1}f(x)=1$. Then $\lim_{x\rightarrow - 1}\sqrt{3 + f(x)}=\sqrt{\lim_{x\rightarrow - 1}(3 + f(x))}=\sqrt{3 + 1}=2$.
Answer:
- $0$
- $1$
- $0$
- $\text{DNE}$
- $2$