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+2x3)4,a=−1

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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+2x3)4,a=−1

falhiblesw · Accepted Answer

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  limx→a−f(x)=limh→0f(a−h)  Right hand limit  limx→a+f(x)=limh→0f(a+h)  So, limit exists at pont a if and only if limx→a−f(x)=limx→a+f(x)= finite  Continuity  A function is said to be continuous at a if and only if  1. Limit exits that is limx→af(x)=limx→a+f(x)= finite  2.  Limit is equal to value of the function at point a that is limx→a−f(x)=limx→a+f(x)=f(a)  Solution:  We have given function f(x)=(x+2x3)4 at points a=-1  Left hand limit  limx→a+f(x)=limx→−1+(x+2x3)4=limh→0((−1+h)+2(−1+h)3)4  =(−1+2(−1)3)4  =(−1−2)4=(−3)4=81  Hence limx→−1+f(x)=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

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Use the definition of continuity and the properties of limits to show that the f

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New Questions in Calculus 1