Q2. - Calculus

Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than

any in the reading.

(a) A function f and a value c such that $\underset{x\to c}{\lim }f\left(x\right)$ happens to be equal to f (c).

(b) A function f and a value c such that $\underset{x\to c}{\lim }f\left(x\right)$ is not equal to f(c).

(c) A function f and a value c such that $\underset{x\to c}{\lim }f\left(x\right)$ exists but f(c) does not exist.

Step-by-Step Solution

Answer

Part (a) $\underset{x\to 2}{\lim }4=4$

Part (b) $\underset{x\to 1}{\lim }2x$

Part (c) $\underset{x\to 1}{\lim }\frac{1}{x-1}$

Part (a) Step 1. Explanation.

Consider the given information, 

Let's take an example to show this statement. The example of a function f  and a value c such that $\underset{x\to c}{\lim }f\left(x\right)$  happen to be equal to $f\left(c\right)$.

That   $\underset{x\to 2}{\lim }4=4$.

 

Here, for $x\to 2^{-}$, the value of the function is $f\left(2^{-}\right)=4$ .

And again, for $x\to 2^{+}$ , the value of the function is   $f\left(2^{+}\right)=4$.

 

Thus, the example is $\underset{x\to 2}{\lim }4=4$.

Part (a) Step 2. Explanation.

Consider the example.

$\underset{x\to 1}{\lim }2x$

Take the left limit of the function.

$\begin{array}{rcl}\underset{x\to 1^{-}}{\lim }2x& =& 2(0.999)\\ & =& 1.998\end{array}$

Take the right limit of the function.

$\begin{array}{rcl}\underset{x\to 1^{+}}{\lim }\left(2x\right)& =& 2\left(1.0001\right)\\ & =& 2.0002\end{array}$

part (c) Step 1. Explanation.

Consider the given information,

Let's take an example to explain the statement. $\underset{x\to 1}{\lim }\frac{1}{x-1}$.

In this case, the limit does not exist if we take the left and right limits.