Limits

**Sanza** · March 20th 2010, 01:23 AM

Prove the following limit of a function using the definition of a limit (where c>0)

$\lim_{x\rightarrow{c}}{\frac{1}{x^3}} = \frac{1}{c^3}$

**Tinyboss** · March 20th 2010, 01:51 PM

You need to give a formula for delta. It will depend on epsilon and c.

**Sanza** · March 23rd 2010, 06:14 AM

In response to the reply,no formula for delta depending on epsilon or c was given.For completeness sake,i will state the definition.

DEFINITION
Let $A \subseteq \mathbb{R}$ , and let c be a cluster point of $A$ . For a function $f : A \rightarrow{\mathbb{R}}$ , a real number $L$ is said to be a limit of $f$ at c if, given any $\epsilon > 0$ there exists a $\delta > 0$ such that if $x \in A$ and $0 < \mid x - c \mid < \delta$ then $\mid f(x) - L \mid < \epsilon$ .

QUESTION
Prove the following limit of a function using the (above) definition where
c > 0
$\lim_{x \rightarrow{c}}{\frac{1}{x^3}} = \frac{1}{c^3}$

**Tinyboss** · March 23rd 2010, 08:24 AM

You need to find a formula for delta, depending on epsilon and c, such that the above definition will be satisfied for any epsilon>0 and any c>0.

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