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July 22, 2014

July 22, 2014

Posted by **Liz** on Wednesday, June 27, 2012 at 1:07pm.

Question: Find the limit L. Then use the å-ä definition to prove the limit is L.

1. lim (2x+5)

x->3

2. lim 3

x->6

Thank you for your anticipated help!!

- Calculus -
**Liz**, Wednesday, June 27, 2012 at 1:08pm**Note**

Please use the Epsilon-Delta Definition.

- Calculus -
**Count Iblis**, Wednesday, June 27, 2012 at 3:51pm1)

f(x) = 2 x + 5

The limit for x to 3 is 11, assuming the function is continuous. To prove this, consider the open interval

I = (11-epsilon, 11+epsilon)

for some arbitrary epsilon

We want to find a delta > 0 such that the function f maps the open interval

J =(3-delta, 3+delta)

into I. That would then prove that for any arbitry epsilon>0 such a delta indeed exists. This means that no matter how small you take epsilon, there is always an interval containing x = 3 that will be mapped by f into the interval of size 2 epsilon containing 11.

If we solve:

2 x + 5 = y

for x, we get:

x = (y-5)/2

If we take y = 11 +/- epsilon, we get:

x = 3 +/- epsilon/2

So, we can take delta = epsilon/2

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