enter each answer as a whole number (like -2, -5, 626, 73772), or a fraction (like 35/4983) or dne for…

enter each answer as a whole number (like -2, -5, 626, 73772), or a fraction (like 35/4983) or dne for undefined (does not exist).\nlim_{x→2^+} (f(x) - 4)/f(x + 1)=\nlim_{x→2^-} f(f(x)+4)=\nlim_{h→0} (f(2 + h)-f(2))/h=

enter each answer as a whole number (like -2, -5, 626, 73772), or a fraction (like 35/4983) or dne for undefined (does not exist).\nlim_{x→2^+} (f(x) - 4)/f(x + 1)=\nlim_{x→2^-} f(f(x)+4)=\nlim_{h→0} (f(2 + h)-f(2))/h=

Answer

Explanation:

Step1: Analyze $\lim_{x\rightarrow2^{+}}\frac{f(x)-4}{f(x + 1)}$

As $x\rightarrow2^{+}$, from the graph, $f(x)\rightarrow3$ and $f(x + 1)$ when $x\rightarrow2^{+}$ means considering $f(3^{+})$ and $f(3^{+})=3$. So $\lim_{x\rightarrow2^{+}}\frac{f(x)-4}{f(x + 1)}=\frac{3 - 4}{3}=-\frac{1}{3}$.

Step2: Analyze $\lim_{x\rightarrow2^{-}}f(f(x)+4)$

As $x\rightarrow2^{-}$, $f(x)\rightarrow3$. Then $f(x)+4\rightarrow7$. From the graph, $f(7)=6$. So $\lim_{x\rightarrow2^{-}}f(f(x)+4)=6$.

Step3: Analyze $\lim_{h\rightarrow0}\frac{f(2 + h)-f(2)}{h}$

This is the definition of the derivative at $x = 2$. The left - hand limit of the slope as $h\rightarrow0^{-}$ and the right - hand limit of the slope as $h\rightarrow0^{+}$ are not equal (the graph has a corner at $x = 2$). So $\lim_{h\rightarrow0}\frac{f(2 + h)-f(2)}{h}$ DNE.

Answer:

$-\frac{1}{3}$ $6$ DNE