Question

$f(x) = \

$$\begin{cases} x + 2 & \\text{if } x \\leq 3 \\\\ 4x - 7 & \\text{if } x > 3 \\end{cases}$$

$

let $f$ be the function given above. which of the following statements are true about $f$?

i. $\lim_{x \to 3} f(x)$ exists.

ii. $f$ is continuous at $x = 3$.

iii. $f$ is differentiable at $x = 3$.

options:

a none

b i only

c ii only

d i and ii only

e i, ii, and iii

Explanation:

Step1: Check Limit Existence (I)

To check if $\lim_{x\to 3} f(x)$ exists, we find the left - hand limit (LHL) and right - hand limit (RHL) as $x\to 3$.

For the left - hand limit, when $x\leq3$, $f(x)=x + 2$. So, $\lim_{x\to 3^{-}}f(x)=\lim_{x\to 3^{-}}(x + 2)$. Substitute $x = 3$ into $x + 2$, we get $3+2 = 5$.

For the right - hand limit, when $x>3$, $f(x)=4x - 7$. So, $\lim_{x\to 3^{+}}f(x)=\lim_{x\to 3^{+}}(4x - 7)$. Substitute $x = 3$ into $4x-7$, we get $4\times3 - 7=12 - 7 = 5$.

Since $\lim_{x\to 3^{-}}f(x)=\lim_{x\to 3^{+}}f(x) = 5$, the limit $\lim_{x\to 3}f(x)$ exists. So statement I is true.

Step2: Check Continuity at $x = 3$ (II)

A function $y = f(x)$ is continuous at $x=a$ if $\lim_{x\to a}f(x)=f(a)$.

We know that $\lim_{x\to 3}f(x) = 5$. Now, find $f(3)$. Since for $x\leq3$, $f(x)=x + 2$, then $f(3)=3 + 2=5$.

Since $\lim_{x\to 3}f(x)=f(3)=5$, the function $f(x)$ is continuous at $x = 3$. So statement II is true.

Step3: Check Differentiability at $x = 3$ (III)

A function $y = f(x)$ is differentiable at $x=a$ if the left - hand derivative (LHD) and right - hand derivative (RHD) at $x = a$ exist and are equal.

The derivative of a function $y = mx + c$ is $y^\prime=m$.

For the left - hand derivative (when $x\leq3$), $f(x)=x + 2$, so $f^\prime(x)=1$ for $x<3$. Then the left - hand derivative at $x = 3$, $f^\prime(3^{-}) = 1$.

For the right - hand derivative (when $x>3$), $f(x)=4x - 7$, so $f^\prime(x)=4$ for $x>3$. Then the right - hand derivative at $x = 3$, $f^\prime(3^{+}) = 4$.

Since $f^\prime(3^{-})=1
eq f^\prime(3^{+}) = 4$, the function $f(x)$ is not differentiable at $x = 3$. So statement III is false.

Answer:

D. I and II only