Calculating limits
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Calculating Limits. AP Calc 2.2. Computing Limits. 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.

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Calculating Limits

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Calculating limits

Calculating Limits

AP Calc 2.2


Computing limits

Computing Limits

  • 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!


Some basic limits to know

Some Basic Limits to Know

  • 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:


Some basic limits to know1

Some Basic Limits to Know

  • Now let’s graph y = x.

    y=x

  • Pick an x value to approach.

  • What is the limit there?

  • So:


Properties of limits

Properties of Limits

  • 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:


Algebraic limits

Algebraic Limits

  • 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…


Algebraic limits1

Algebraic Limits

  • Ex #1:

  • Ex #2:

  • Ex #3:

  • Ok, Ok… let’s get to some good ones!


Algebraic limits2

Algebraic Limits

  • Ex #4:

  • Ex #5:

  • Ex #6:

  • Ex #7:


One sided limits

One Sided Limits

  • If there is no graph available, plug in values approaching the given x – value.

  • Ex:

  • Let’s make a table of values:


One sided limits1

One Sided Limits

  • 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 ∞


Limits with radicals

Limits with Radicals

  • 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.


Piecewise functions

Piecewise Functions

  • Let’s look at

  • Find the following limits:


Homework

Homework

  • Pg. 121 # 1 – 31 odd, #2, and #37


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