Find out the answer to "Continuity and limts: Show a function is continuous..."

nagasenaz

Answered question

2021-10-29

Continuity and limts: Show a function is continuous at a

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

Sadie Eaton

Skilled2021-10-30Added 104 answers

Show a function is continuous at a

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

We have

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

f (x) = {(x + 2 x^{3})}^{4}

at a=-1 to be continous
if, L.H.L=R.H.L=Function at the given point
Now,

lim_{x \to - 1} f (x) = lim_{x \to - 1} {(x + 2 x^{3})}^{4} = {(lim_{x \to - 1} x + 2 {(lim_{x \to - 1} x)}^{3})}^{4}

\Rightarrow lim_{x \to - 1} f (x) = {(- 1 + 2 {(- 1)}^{3})}^{4} = {(- 1 - 2)}^{4} = {(- 3)}^{4}

\Rightarrow lim_{x \to - 1} f (x) = 81

∴ L . H . L = lim_{x \to - 1} f (x) = 81

Then,

lim_{x \to 1} f (x) = {(1 + 2 {(1)}^{3})}^{5} = {(1 + 2)}^{4} = {(3)}^{4}

∴ R . H . L = lim_{x \to 1} f (x) = 81

Now, for f(-1)

\Rightarrow f (- 1) = {((- 1) + 2 {(- 1)}^{3})}^{4} = {(- 1 - 2)}^{4} = {(- 3)}^{4}

\Rightarrow f (- 1) = 81

Here,

L . H . L = R . H . L = f (- 1)

Hence, f(x) is continuous at a=-1

Jeffrey Jordon

Expert2022-08-30Added 2605 answers

Answer is given below (on video)

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