the graphs of f and g are given. use them to evaluate each limit, if it exists. (if an answer does not…

the graphs of f and g are given. use them to evaluate each limit, if it exists. (if an answer does not exist, enter dne.) (a) $lim_{x\rightarrow2}f(x)+g(x)$ (b) $lim_{x\rightarrow1}f(x)+g(x)$ (c) $lim_{x\rightarrow0}f(x)g(x)$ (d) $lim_{x\rightarrow - 1}\frac{f(x)}{g(x)}$ (e) $lim_{x\rightarrow2}x^{3}f(x)$ (f) $lim_{x\rightarrow1}sqrt{3 + f(x)}$ resources read it watch it

the graphs of f and g are given. use them to evaluate each limit, if it exists. (if an answer does not exist, enter dne.) (a) $lim_{x\rightarrow2}f(x)+g(x)$ (b) $lim_{x\rightarrow1}f(x)+g(x)$ (c) $lim_{x\rightarrow0}f(x)g(x)$ (d) $lim_{x\rightarrow - 1}\frac{f(x)}{g(x)}$ (e) $lim_{x\rightarrow2}x^{3}f(x)$ (f) $lim_{x\rightarrow1}sqrt{3 + f(x)}$ resources read it watch it

Answer

Explanation:

Step1: Recall limit - sum rule

$\lim_{x\rightarrow a}[f(x)+g(x)]=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$ (if both $\lim_{x\rightarrow a}f(x)$ and $\lim_{x\rightarrow a}g(x)$ exist).

Step2: Recall limit - product rule

$\lim_{x\rightarrow a}[f(x)g(x)]=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow a}g(x)$ (if both $\lim_{x\rightarrow a}f(x)$ and $\lim_{x\rightarrow a}g(x)$ exist).

Step3: Recall limit - quotient rule

$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a}g(x)}$, provided $\lim_{x\rightarrow a}g(x)\neq0$.

Step4: Recall limit - constant - multiple rule

$\lim_{x\rightarrow a}[cf(x)] = c\lim_{x\rightarrow a}f(x)$ for a constant $c$. Here, for $\lim_{x\rightarrow 2}[x^{3}f(x)]$, we use $\lim_{x\rightarrow a}[u(x)v(x)]=\lim_{x\rightarrow a}u(x)\cdot\lim_{x\rightarrow a}v(x)$ with $u(x)=x^{3}$ and $v(x)=f(x)$.

Step5: Recall limit - composition rule for root functions

$\lim_{x\rightarrow a}\sqrt{u(x)}=\sqrt{\lim_{x\rightarrow a}u(x)}$ when $\lim_{x\rightarrow a}u(x)\geq0$.

(a)

  • First, find $\lim_{x\rightarrow 2}f(x)$ and $\lim_{x\rightarrow 2}g(x)$ from the graphs.
    • From the graph of $y = f(x)$, $\lim_{x\rightarrow 2}f(x)=1$.
    • From the graph of $y = g(x)$, $\lim_{x\rightarrow 2}g(x)=1$.
    • Then $\lim_{x\rightarrow 2}[f(x)+g(x)]=\lim_{x\rightarrow 2}f(x)+\lim_{x\rightarrow 2}g(x)=1 + 1=2$.

(b)

  • $\lim_{x\rightarrow 1}f(x)=2$ and $\lim_{x\rightarrow 1}g(x)=1$.
    • $\lim_{x\rightarrow 1}[f(x)+g(x)]=\lim_{x\rightarrow 1}f(x)+\lim_{x\rightarrow 1}g(x)=2 + 1=3$.

(c)

  • $\lim_{x\rightarrow 0}f(x)= - 1$ and $\lim_{x\rightarrow 0}g(x)=0$.
    • $\lim_{x\rightarrow 0}[f(x)g(x)]=\lim_{x\rightarrow 0}f(x)\cdot\lim_{x\rightarrow 0}g(x)=(-1)\times0 = 0$.

(d)

  • $\lim_{x\rightarrow - 1}f(x)= - 1$ and $\lim_{x\rightarrow - 1}g(x)=0$. Since $\lim_{x\rightarrow - 1}g(x)=0$, $\lim_{x\rightarrow - 1}\frac{f(x)}{g(x)}$ does not exist (DNE).

(e)

  • $\lim_{x\rightarrow 2}x^{3}=2^{3}=8$ and $\lim_{x\rightarrow 2}f(x)=1$.
    • $\lim_{x\rightarrow 2}[x^{3}f(x)]=\lim_{x\rightarrow 2}x^{3}\cdot\lim_{x\rightarrow 2}f(x)=8\times1 = 8$.

(f)

  • $\lim_{x\rightarrow 1}f(x)=2$. Then $\lim_{x\rightarrow 1}(3 + f(x))=3+\lim_{x\rightarrow 1}f(x)=3 + 2=5$.
    • $\lim_{x\rightarrow 1}\sqrt{3 + f(x)}=\sqrt{\lim_{x\rightarrow 1}(3 + f(x))}=\sqrt{5}$.

Answer:

(a) 2 (b) 3 (c) 0 (d) DNE (e) 8 (f) $\sqrt{5}$