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

Anonym 2021-10-26 Answered
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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falhiblesw
Answered 2021-10-27 Author has 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
limxaf(x)=limh0f(ah)
Right hand limit
limxa+f(x)=limh0f(a+h)
So, limit exists at pont a if and only if limxaf(x)=limxa+f(x)= finite
Continuity
A function is said to be continuous at a if and only if
1. Limit exits that is limxaf(x)=limxa+f(x)= finite
2. Limit is equal to value of the function at point a that is limxaf(x)=limxa+f(x)=f(a)
Solution:
We have given function f(x)=(x+2x3)4 at points a=-1
Left hand limit
limxa+f(x)=limx1+(x+2x3)4=limh0((1+h)+2(1+h)3)4
=(1+2(1)3)4
=(12)4=(3)4=81
Hence limx1+f(x)=81
So limit exists
Now, f(1)=(1+2(1)3)4
=(12)4=(3)4=81
Therefore,
Given function is continuous at x=-1
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Jeffrey Jordon
Answered 2022-06-24 Author has 2262 answers

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