Calculus Solution Exercise 1.1

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Q.01

For the function g graphed in …. find Solution

Solution:

(a) $\operatorname{lim}_{x \rightarrow 0^{-}} g(x)=3$

This is because as $x$ approaches o from the left side, $g(x)$ approaches 3.

(b) $\quad \lim _{x \rightarrow 0^{+}} g(x)=3$

From the graph, we can see that as $x$ approaches o from the right side, $g(x)$ approaches 3.

that is the left-hand limit and right-hand limit both are equal to 3, therefore $\operatorname{lim}_{x \rightarrow 0} g(x)=3$
(d) $g(0)=3$ , the value of $g(x)$ at $x=0$ which is 3 .

Q.02

For which function $G(x)$ graphed…. figure.

Solution:

$\text { (a) } \operatorname{lim}_{x \rightarrow 0^{-}} G(x)$

It is clear from the graph(see book), as $x$ approaches to 0 from the left sides, the $G(x)$ approaches to 0

$\therefore \quad \operatorname{lim}_{x \rightarrow 0^{-}} G(x)=0$

(b)

$\lim _{x \rightarrow 0^{+}} G(x)=0$

From the figure(see book), we see that as $x$ approaches 0 from the right side, $G(x)$ approaches 0.
(c)

$\begin{aligned} & \operatorname{lim}_{x \rightarrow 0} G(x)\\ \text {since} \quad &\operatorname{lim}_{x \rightarrow 0^{+}} G(x)=0=\lim _{x \rightarrow 0^{-}}} G(x) \\ \\ & \therefore \lim _{x \rightarrow 0} G(x)=0 \end{aligned}$

(d) $G(0)=0$ , the value of $G$ is zero at $x=0$ (see the graph in the book)

Q.03 For the function $f$ graphed….. find…

Solution:

(a)

$\operatorname{lim}_{x \rightarrow 3^{-}} f(x)=-1$

We see from the graph(see book), that as $x$ approaches 3 from the left side, the $f(x)$ approaches -1.

(b) $\quad \lim _{x \rightarrow 3^{+}} f(x)=3$

As $x$ approaches 3 from the right sides, $f(x)$ approaches 3.

(c)

$\begin{aligned} & \operatorname{lim}_{x \rightarrow 3} f(x) \\ \\ & \text { Since } \operatorname{lim}_{x \rightarrow 3^{-}} f(x) \neq \operatorname{lim}_{x \rightarrow 3^{+}} f f(x) \end{aligned}$

that is the left-hand limit is not equal to the right-hand limit, therefore $\lim _{x \rightarrow 3} f(x)$ does not exist.

(d) $f(3)=1$ , the value of $f$ at $x=3$ .

Q.04 For which function $f$ …. find

Solution:

(a) $\lim _{x \rightarrow 2^{-}} f(x)=2$

As $x$ approaches to 2 from the left side, $f(x)$ approaches to 2 ,
(b) $\quad \lim _{x \rightarrow 2^{+}} f(x)=0$
As $x$ approaches 2 from the right side, $f(x)$ approaches 0.

(c)

$\text{ Since } \operatorname{lim}_{x \rightarrow 2^{-}} f(x) \neq \operatorname{lim}_{x \rightarrow 2^{+}} f(x)$

that is the left-hand limit and right-hand limit are not same, therefore $\lim _{x \rightarrow 3} f(x)$ does not exist.

(d) $f(2)=2$ the value of $f$ at $x=2$ .

Q.01

Q.02

Q.03

For the function graphed….. find…

Q.04

For which function …. find

For the function $f$ graphed….. find…

For which function $f$ …. find