Theorems on Limits

If

1)	(limit of sum or difference = sum or difference of limits)
2)	(limit of product = product of limits
3) , M ! 0	(limit of quotient = quotient of limits)
4)	(limit of scalar product = scalar times the limit)

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Examples

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Limit Theorem on

This function has a vertical asymptote at x = a.

What the y values do as x d a depends on whether n is even or odd.

odd

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here, n is odd (n = 1)

, while

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even

here, here, n is even (n = 2)

, and

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If f(x) = if n is odd then and (graph) if n is even then and (graph)

In the case of f(x) =, use the theorem but multiply the limit by to adjust the sign of the limit according to the sign of . If < 0, (negative) the sign of the limit will change.

Say we want . When we factor, we get .

Since n is odd (7), the left limit should be - º, but it's multiplied by ,

so the left limit will be z and the right limit will be - º.

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Limits at Infinity

note: Since it is true that "the larger the divisor, the smaller the quotient", division by infinity equals zero.

note: Since it is true that "the smaller the divisor, the larger the quotient", division by zero equals infinity.

If a is a scalar, then:

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algorithm for finding when f(x) is a rational function

If f(x) is a rational function (fraction), to find the limit as x d º divide each term in the fraction by x to the degree (highest power) of the fraction. Then set x = º, and evaluate the limit.

fraction degree = 2, we divide by x ² .

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fraction degree = 3, we divide by x ³ .

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sometimes it is not obvious what the degree of the fraction is:

here, the degree is 1, however, in the numerator, x ¹ is expressed as since we have to divide within the radical.

Adjust the sign ( + or - ) when you find the limit as x d - º .

note: the limit as x d | º | for a rational function is the value of the horizontal asymptote since it describes the behavior of the function's curve at the extremes of the x-axis. The horizontal asymptote for the first function above is the line y = ½, for the second y = 0.

algorithm for finding when f(x) is not rational

Since infinity comes in different sizes (some are bounded, others are not) the expression is indeterminate. But, we can divide by anything other than 0, so we divide each term in the expression by x to get:

, now we rationalize the numerator to get .

Other cases of

For many functions just look at the graph to know

f(x)
e - x ,	0
e x ,	º
log a x , 0 < a < 1	º
log a x , a > 1	º

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Practice

A/ THEOREM ON : Find the limits

1.	a)	b)	c)
2.	a)	b)	c)
3.	a)	b)	c)
4.	a)	b)	c)

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B/ limits at infinity

1. find if f(x) =

a)	b)	c)	d)
e) -2log 3 (x + 5)		f) e 3 x + 2 - 5

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2. find

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Solutions

A/ THEOREM ON : Find the limits

1.	a) n = 2, left lim = - º	b) n = 2, right lim = - º	c) DNE ( a )
2.	a) n = 7, left lim = - º	b) n = 7, right lim = º	c) DNE ( a )
3.	a) 3x 2 m 0; n = 2 both sides to º	b) 3x 2 m 0; n = 2 both sides to º	c) DNE ( a )
4.	a) 2x 2 m 0; n = 1 x - 2< 0 if x at -1 left limit is º	b) right limit is - º	c) DNE ( a )

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B/ limits at infinity

1. find if f(x) =

a)	b)	c)
e) – 2log 3 (x + 5) = – º	d)	f) e 3 x + 2 – 5 = º

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2. find

As x d - º, y d - 3, the horizontal asymptote.

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