Notes ­ Limits Day 1.notebook

May 22, 2015

Chapter 12 Limits ­ An Intro to Calculus  (12.1,12.2)

Limit ­ fundamental concept of Calculus If f(x) becomes arbitrarily close to a unique #L as x  approaches c from either side, the limit of f(x) as x  approaches c is L.

Can use a table to estimate the limits: 1.

2.    

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The existence or nonexistence of f(x) when x = c has no  bearing on the existence of the limit of f(x) as x approaches c

Can use a graph to find the limits:

1.

Limit of a Piecewise function: 2.      

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Notes ­ Limits Day 1.notebook

May 22, 2015

A Few Properties: Limit of a constant = constant Limit of a sum/diff = the sum/difference of limits Limit of a product/quotient = the product or quotient of the limits

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Notes ­ Limits Day 1.notebook

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Techniques for Evaluating Limits: 1.  Direct Substitution:  polynomial function   rational function such that q(c) ≠ 0

ex.

ex.

2.  Dividing Out Tecnique:  Factor and simplify, then use direct     substitution

ex.

ex.        

  

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Notes ­ Limits Day 1.notebook

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3.  Rationalizing Technique: (Rationalize the numerator)

ex.  

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4.  One­Sided Limits with Piecewise Functions ***If f is a function and c and L are real #s, then   if  and only if both the left and right limits exist and are equal to L.

ex.

ex.

ex.    

    

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Limits that fail to exist: 1.  approaches two different values from the left and right 2.  unbounded behavior (vertical asymptotes) ex.    

3.  oscillating behavior ­­ bounces between two fixed values ex.    

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Notes ­ Limits Day 1.notebook

May 22, 2015

Homework: pg.  813­814 #7­21 odd  (do not have to construct table by hand, use  calculator)

pg. 824­825 #1­9 odd, 13­23 odd, 33­37 odd

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Notes ­ Limits Day 1.notebook

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