infinite limits. limits what is calculus? what are limits? evaluating limits –graphically...
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INFINITE LIMITSINFINITE LIMITS
LIMITSLIMITS
• What is Calculus?• What are Limits?• Evaluating Limits
– Graphically– Numerically– Analytically
• What is Continuity?• Infinite Limits
This presentation
- When the value of the function grows without bounds
- When x approaches
Infinite LimitsThe statement means that the function
grows positively without bounds as x approaches c.
)(lim xfcx
)(lim xfcx
The statement means that the function
grows negatively without bounds as x approaches c.
The equals signs in the statements above do not mean that the limits exist. On the contrary, it says that the limits fail by demonstrating unbounded behavior as x approaches c.
Important note:
and
)(lim xfcx
)(lim xfcx
One sided limits can also be infinite:
Infinite Limits & Rational Functions
• Infinite limits occur at vertical asymptotes.
• Rational functions that cannot be fully simplified generate vertical asymptotes.
• We will limit (no pun intended) our study of infinite limits to rational functions.
For each example, 1. Identify any vertical asymptotes (be sure to simplify the function first
to discount any point discontinuities).2. Graph the function and observe the behavior of the function as it
approaches these x = c values from both directions. Does it grow without bound positively or negatively?
Infinite Limits: Examples
1.
2.
3.
221
)(
x
xf
4
82)(
2
2
x
xxxf
xxf cot)(
VA @ x = -2:
)(lim2
xfx
)(lim2
xfx
Hole @ x = 2:
VA @ x = 1:
)(lim1
xfx
)(lim1
xfx
Infinitely many VA @ :nx
)(lim xf
cx
)(lim xf
cxFor ea VA, x = c, ,
5.1)(lim2
xfx
(Removable point discontinuity)
Evaluating Limits When x Approaches
)(lim xfx is asking about the right hand behavior of the function
)(lim xfx is asking about the left hand behavior of the function
32)( 3 xxxf 132
)(
x
xxf
Examples:
1. 2.
32
)(lim
xfx
)(lim xf
x
)(lim xfx 3
2)(lim
xf
x
This is an odd polynomial witha positive leading coefficient. So the RH behavior And the LH behavior
Therefore:
This is a rational function with ahorizontal asymptote at y = 2/3.
Therefore:
• Infinite limits are written as
– The equals sign is misleading since the limit does not exist.
– Infinite limits generally occur at the vertical asymptotes of rational functions. The function may grow positively or negatively on either side of the asymptote.
• When asking for a limit as x , i.e.,
– Check the end behavior of the function.
Infinite Limits: Summary
)(lim xf
cx
)(lim xfx