Q. 2E - 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 $f\left(c\right)$.

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

(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

(a). $\underset{x\to 1}{\lim }2=2$

(b). $\underset{x\to 1}{\lim }x$

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

part(a) Step 1: Given information

One example of a function $f$ and a value $c$ such that $\lim _{x \rightarrow c} f(x)$ happens to be equal to $f(c)$ is given as below:

$\lim _{x \rightarrow 1} 2=2$

part(a) Step 2: Simplification

$\lim _{x \rightarrow 1} 2=2$

Here, for $x \rightarrow 1^{-}$, the value of the function is $f\left(1^{-}\right)=2$.

Again, for $x \rightarrow 1^{*}$, the value of the function is $f\left(1^{+}\right)=2$.

part(b) Step 1: Given information

One example of a function $f$ and a value $c$ such that $\lim _{x \rightarrow \mathbb{c}} f(x)$ is not equal to $f(c)$ is given as below:

$\lim _{x \rightarrow 1} x$

part(b) Step 2: Simplification

$\lim _{x \rightarrow 1} x$

Here, for $x \rightarrow 1^{-}$, the value of the function is $f\left(1^{-}\right)=0.999$.

Again, for $x \rightarrow 1^{+}$, the value of the function is $f\left(1^{*}\right)=1.0001$.

part(c) Step 1: Given information

One example of a function $f$ and a value $c$ such that $\lim _{x \rightarrow c} f(x)$ exist but $f(c)$ does not exist is given as belowr:

$\lim _{x \rightarrow 1} \frac{1}{x-1}$

part(c) Step 2: Simplification

$\lim _{x \rightarrow 1} \frac{1}{x-1}$

Here, for $x \rightarrow 1^{-}$and $x \rightarrow 1^{*}$, the value of the function exists but $f(1)$ does not exists.