20. let f be the function given above. which of the following statements are true about f? i. lim x→3 f(x)…

Explanation:

Step1: Calculate left - hand limit

For $x\leq3$, $f(x)=x + 2$. So, $\lim_{x\rightarrow3^{-}}f(x)=\lim_{x\rightarrow3^{-}}(x + 2)=3+2=5$.

Step2: Calculate right - hand limit

For $x>3$, $f(x)=4x - 7$. So, $\lim_{x\rightarrow3^{+}}f(x)=\lim_{x\rightarrow3^{+}}(4x - 7)=4\times3-7=12 - 7 = 5$. Since $\lim_{x\rightarrow3^{-}}f(x)=\lim_{x\rightarrow3^{+}}f(x)=5$, $\lim_{x\rightarrow3}f(x)$ exists.

Step3: Check continuity

$f(3)=3 + 2=5$. Since $\lim_{x\rightarrow3}f(x)=5$ and $f(3)=5$, the function $f(x)$ is continuous at $x = 3$.

Step4: Check differentiability

The derivative of $y=x + 2$ is $y^\prime=1$ for $x\leq3$. The derivative of $y = 4x-7$ is $y^\prime=4$ for $x>3$. The left - hand derivative at $x = 3$ is $1$ and the right - hand derivative at $x = 3$ is $4$. Since the left - hand derivative and the right - hand derivative are not equal at $x = 3$, the function is not differentiable at $x = 3$.

Answer:

D. I and II only