Calculus: Use limits to show that the function is continuous at the given number a.
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# Calculus: Use limits to show that the function is continuous at the given number a.

[From: ] [author: ] [Date: 11-06-05] [Hit: ]
we see that f(x) is continuous at x = -1.I hope this helps!......
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+2x^3)^4, a=-1.

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If f(x) is continuous at x = a, then:
lim (x-->a-) f(x) = lim (x-->a+) f(x) = f(a).

Since:
(i) lim (x-->-1-) f(x)
= lim (x-->-1-) (x + 2x^3)^4
= [-1 + 2(-1)]^4
= 81
(ii) lim (x-->-1+) f(x)
= lim (x-->-1+) (x + 2x^3)^4
= [-1 + 2(-1)]^4
= 81
(iii) f(-1) = [-1 + 2(-1)]^4 = 81
==> lim (x-->-1-) f(x) = lim (x-->-1+) f(x) = f(-1),

we see that f(x) is continuous at x = -1.

I hope this helps!

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the limit does not exist
1
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