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I have a question in my text that I don't quite know what to do with:

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

so far I have gotten:

  (lim x + 2 lim x^3)^4  
   x->a      x->a
  1. is this correct so far?
  2. if so, what is my next move?

Thanks in advance!!!! (loving this site by the way :-))

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Try to make your question self contained, we do not know what you mean by law 1 and 3 :). – sxd Feb 7 '13 at 2:42
I imagine one of the laws is just the fact that it's legal to pull a constant in front of the limit. The other is just the fact that you're allowed to add two continuous, smooth functions under one limit and split it into two limits. – Joe Feb 7 '13 at 2:45
It is clear that $\lim_{x\to -1}x=-1$. The rest is taken care of by your laws, limit of a sum is the sum of the limits, limit of a product is the product of the limits. – André Nicolas Feb 7 '13 at 2:48
Thank You Andre'!!! – codenamejupiterx Feb 7 '13 at 3:55
up vote 1 down vote accepted

Assuming, as suggested that the laws you are using, applied to continuous functions, are

  • the limit of the sum is the sum of the limits, and
  • the limit of the product is the product of the limits (to take care of the "pushing the limit through a power, by considering it as a product limits)

then what you've done is fine.

You simply need to replace $a$ with $-1$ and evaluate the two limits as $x \to -1$, sum them, then take the sum to a power of $4$.

The result will give you $\lim_{x\to a} f(x)$, given $f(x)$ and given $a = -1$. With that result, you can confirm that indeed, the function is continuous at the given $a$.

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Thank You amWhy! – codenamejupiterx Feb 7 '13 at 3:54
You're welcome! – amWhy Feb 7 '13 at 3:57

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