# Warm Up by yurtgc548

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```									                  Warm Up
• Find the limit as x approaches 1 for the
following functions
• f(x) = 3
• f(x) = x
• f(x) = 3x +5
• f(x) = x2 +4x – 2
• f(x) = (x3 – 1)/(x – 1)
Alternatives to Direct
Substitution
September 10, 2010
Objective
• SWBAT use an alternative to direct
substitution to evaluate the limit of a function
Vocabulary
Rationalize   Multiplying by a conjugate to
expression

Conjugate     Two identical binomials that have
opposite signs
Alternative Strategies
• When evaluating using direct substitution
produced 0 in the denominator, direct
substitution will not work to find a limit.
• If you have a polynomial in the numerator,
factor the polynomial.
• If the numerator and denominator now share
a factor, cancel it out and try direct
substitution again.
Alternative Strategies
• If you have a radical (square root) in the
numerator, multiply the numerator and
denominator by the conjugate of the
numerator.
• This should eliminate the radical and change
the denominator so you can use direct
substitution.
Example
x3  1
1. lim
x 1 x  1
Example
x2  x  6
2. lim
x 3    x 3
Example
x 1  1
3. lim
x 0      x
Practice
x  3x
2
x 3 1
1. lim                             lim
1. x 1
x 0   x                              x 1

x                              2 x  2
   2. lim 2
x 1 x  x                2. lim
x 0      x

2x 2  x  3                    x 1  2
3.   lim
    3. lim
x 1     x 1                x 3     x 3

                             
Practice
x3  8                          x5 5
lim                          lim
x 2 x  2
x 0      x

2 x                            x53
lim 2                         lim
x 2 x  4                    x 4    x4
                        
x2  x  6
lim
x 3   x2  9
                         
x 2  5x  4
lim 2
x 4 x  2x  8


```
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