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Calculating Limits

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Calculating Limits

AP Calc 2.2

- As we saw in the last section, graphs can be deceiving!
- Think of holes shown (or not shown) on the calculator.

- There is also the possibility that the graph is not readily available.
- So we need to know how to calculate limits without pictures!

- Let’s come up with a couple of limits on our own…
- Let’s draw a graph of a constant:
k

- Pick an x – value to approach.
- What is the limit there?
- So:

- Now let’s graph y = x.
y=x

- Pick an x value to approach.
- What is the limit there?
- So:

- Please turn to page 115 in your books.
- Let’s look at the box first…
- BIG FACT:
- A constant factor can be mover through the limit symbol!
- Ex:

- We will be using the 4 step system to find limits:
- Try substituting the given x value first.
- Simplify/Factor Completely
- Try substitution again.
- Sometimes limits DNE!
- Let’s do some examples…

- Ex #1:
- Ex #2:
- Ex #3:
- Ok, Ok… let’s get to some good ones!

- Ex #4:
- Ex #5:
- Ex #6:
- Ex #7:

- If there is no graph available, plug in values approaching the given x – value.
- Ex:
- Let’s make a table of values:

- If there is no graph available, plug in values approaching the given x – value.
- Ex:
- Let’s make a table of values:
So the limit as x approaches

from the left is ∞

- Let’s look at this one:
- Can we just plug in 1?
- How do you simplify this?
- Rationalize! (aka use the conjugate)
- One tip before we do this.
- Don’t do the multiplication of the “unrelated” parts.
- This will make sense in a minute.

- Let’s look at
- Find the following limits:

- Pg. 121 # 1 – 31 odd, #2, and #37