"Use the definition of continuity and the properties..." was answered by our community

Lipossig

Answered question

2021-10-16

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

f (x) = {(x + 2 x^{5})}^{4}, a = - 1

Nola Robson

Skilled2021-10-17Added 94 answers

A function is continuous at a number a if

lim_{x \to a} f (x) = f (a)

From the given data:

f (x) = {(x + 2 x^{5})}^{4}

a = - 1

Now,

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

Substituting the limit to the equation:

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

lim_{x \to^{- 1}} f (x) = - 3 \cdot - 3 \cdot - 3 \cdot - 3 = 81

Now,

f (- 1) = {((- 1) + 2 {(- 1)}^{5})}^{4}

f (- 1) = {(- 3)}^{4}

f (- 1) = 81

Thus,

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

So, the function f is continuous at x=-1

Jeffrey Jordon

Expert2022-06-24Added 2605 answers

Answer is given below (on video)

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