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Limits of Functions
Eric Hoffman
Calculus
PLHS
Sept. 2007
Key Topics
L is the limit of the function of f as x approaches a, written:
)(lim xfLax
if the values of f(x) approach the unique number L as x approaches a from either direction
Look at picture on pg. 95 of book
Key Topics• Quadratic Function : the limit of a quadratic function
is of the form CBaAaCBxAxax
22 )(lim
from this we can see that the limit as x approaches a of a quadratic function f is just the value f(a)
Ex. Let f(x) = 3x2 – 2x + 3
243)3(2)3(3323lim22
3
xx
x
Functions that have the property
are called continuous functions
)()(lim afxfax
Key Topics• Limit of a function that is not continuous: if a
function is not continuous it basically means that the function has an asymptote
• Ex. Let this function is undefined at
x=3, so if we want to find the limit of this function at x=3 we can’t just plug 3 in for “a”. This is because the function is not continuous at x=3
To solve we must factor out the “offending” factor
3
6)(
2
x
xxxf
Key Topics
3
6)(
2
x
xxxf 3
6lim)(lim
2
33
x
xxxf
xx
3
)2)(3(lim
3
x
xxx
)2(3
3lim
3
xx
xx
)2(lim3
xx
5)23(lim3
x
Key Topics• Limits that don’t exist:
If we factor the numerator we notice that (x-3) is not a factor, thus we can’t cancel anything out
As x approaches 3 the numerator approaches 6 and the denominator approaches 0 thus the quotient “blows up”
3
6lim
2
3
x
xxx
3
)2)(3(lim
3
x
xxx
Key Topics• Properties of limits: let f and g be functions
for which and
and let c be any real number. Then:
Lxfax
)(lim Mxgax
)(lim
MLxgxfxgfaxaxax
)(lim)(lim))((lim
cLxfcxcfaxax
)]([lim))((lim
MLxgxfxfgaxaxax
)(lim)(lim))((lim
MLxgxfxgfaxaxax
/)(lim/)(lim))(/(lim
Provided m≠0
Applying the Properties of Limits
• Find 623lim2
3
xx
x
623lim2
3
xx
x)6(lim2lim3lim
33
2
3
xxxxx
)6()3(2)3(3 2 27
Key Topics
3
81lim
2
9
x
xx 3
)9)(9(lim9
x
xxx
3
)9)(3)(3(lim9
x
xxxx
)9)(3(lim9
xxx
108)18)(6(
Key Topics
• Homework pg. 100 3-24,multiples of 3
3,6,9…
8 problems!!
