Calculus

I have two similar problems that I need help completing. Please show all your work.

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!!

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  1. **Note**
    Please use the Epsilon-Delta Definition.

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  2. 1)

    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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