Help with Limits?

Question

Trying to make sure I have these correct for the following problems. Thank you for the help.

1. Limit [as x approaches 1] of: x^1/(1-x)

2. Limit [as t approaches 0] of: e^(qt - pt)/t

3. Limit [as t approaches pi (3.14)] of: sin(t)/ (t^2- pi^2) ------ pi = 3.14

4. Limit [as x approaches infinity] of: ln(ln(x))/ln(x)

Thank you again for your insight.

Answer 1

Rule of thumb for limits: first, try substituting the value x approaches into the function, and you'll either end up with:
(1) some value
(2) some value divided by 0, or
(3) 0/0
(4) some value divided by infinity
(5) infinity divided by some value
(6) infinity/infinity


If (1) is true, then the answer will be just that value.
If (2) is true, the limit does not exist.
If (3) is true, the limit may exist and you'll have to algebraically determine what the limit is.
If (4) is true, the value will be zero.
If (5) is true, the value will be infinity (or "does not exist" may be valid too)
If (6) is true, the limit may exist.

Cases #3 and #6 involve using L'Hospital's Rule, where you take the derivative of the numerator and denominator and calculate the new limit.
Let's try these tricks on each question.
1. Lim (x --> 1, x^1/(1-x))

In this case, substituting 1 into the function yields 1/(1-1) = 1/0, so the limit doesn't exist.

2. Lim (t ---> 0, e^(qt - pt)/t)

Substitute 0, and you get e^(0-0)/0, or 1/0, so the limit doesn't exist.

3. Lim (t ---> pi, sin(t)/(t^2 - pi^2))

Substitute pi to get sin(pi)/(pi^2 - pi^2), and we get 0/0, so we have to proceed.

Use L'Hospital's rule (it should be in your notes somewhere), where you take the derivative of the numerator and denominator of the fraction to get a new limit. So the new limit becomes:

Lim (t --> pi, cos(t)/(2t)) and at this point we substitute pi, to get
cos(pi) / (2*pi) = -1/2*pi, which is our answer.

4. Lim (t --> infinity, ln(ln(x))/ln(x))
This is an infinity/infinity case, so we use L'Hospital's Rule.

Lim (t --> infinity, (1/ln(x) * 1/x) / (1/x) ) )

That meshes into

Lim (t --> infinity, 1/ln(x))

Which becomes a 1/infinity case, so the answer is 0.

Answer 2

1. Limit [as x approaches 1] of: x^1/(1-x)
does not exist

2. Limit [as t approaches 0] of: e^(qt - pt)/t
does not exist

3. Limit [as t approaches pi (3.14)] of: sin(t)/ (t^2- pi^2) ------ pi = 3.14
=cos(pi)/2pi

4. Limit [as x approaches infinity] of: ln(ln(x))/ln(x)
=0

.

Answer 3

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