Anonym

2021-10-26

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.

$f\left(x\right)={(x+2{x}^{3})}^{4},a=-1$

falhiblesw

Skilled2021-10-27Added 97 answers

Limits

Limit of the function at point a exists if and only if left hand limit and right hand limit are equal and exists finitely at point a.

Left hand limit

$\underset{x\to {a}^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(a-h)$

Right hand limit

$\underset{x\to {a}^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(a+h)$

So, limit exists at pont a if and only if$\underset{x\to {a}^{-}}{lim}f\left(x\right)=\underset{x\to {a}^{+}}{lim}f\left(x\right)=$ finite

Continuity

A function is said to be continuous at a if and only if

1. Limit exits that is$\underset{x\to a}{lim}f\left(x\right)=\underset{x\to {a}^{+}}{lim}f\left(x\right)=$ finite

2. Limit is equal to value of the function at point a that is$\underset{x\to {a}^{-}}{lim}f\left(x\right)=\underset{x\to {a}^{+}}{lim}f\left(x\right)=f\left(a\right)$

Solution:

We have given function$f\left(x\right)={(x+2{x}^{3})}^{4}$ at points a=-1

Left hand limit

$\underset{x\to {a}^{+}}{lim}f\left(x\right)=\underset{x\to -{1}^{+}}{lim}{(x+2{x}^{3})}^{4}=\underset{h\to 0}{lim}{((-1+h)+2{(-1+h)}^{3})}^{4}$

$={(-1+2{(-1)}^{3})}^{4}$

$={(-1-2)}^{4}={(-3)}^{4}=81$

Hence$\underset{x\to -{1}^{+}}{lim}f\left(x\right)=81$

So limit exists

Now,$f(-1)={(-1+2{(-1)}^{3})}^{4}$

$={(-1-2)}^{4}={(-3)}^{4}=81$

Therefore,

Given function is continuous at x=-1

Limit of the function at point a exists if and only if left hand limit and right hand limit are equal and exists finitely at point a.

Left hand limit

Right hand limit

So, limit exists at pont a if and only if

Continuity

A function is said to be continuous at a if and only if

1. Limit exits that is

2. Limit is equal to value of the function at point a that is

Solution:

We have given function

Left hand limit

Hence

So limit exists

Now,

Therefore,

Given function is continuous at x=-1

Jeffrey Jordon

Expert2022-06-24Added 2575 answers