Calculus Cheat Sheet Limits

Calculus Cheat Sheet

Limits Definitions Precise Definition : We say lim  f ( x ) = L if  Limit at Infinity : We say lim  f ( x ) = L if we  x ®a

 x ®¥

> 0 there is a d  > 0 such that whenever 0 < x - a < d  then  f ( x ) - L < e  .

can make  f ( x ) as close to L to L as we want by

“Working” Definition : We say lim  f ( x ) = L

There is a similar definition for  lim  f ( x ) = L

if we can make  f ( x ) as close to L to L as we want

except we require x require x large and negative.

 by taking x taking x sufficiently close to a (on either side of a of a) without letting  x = a .

Infinite Limit : We say lim  f ( x ) = ¥ if we

for every

e 

 x ®a

Right hand limit : lim+  f ( x ) = L . This has  x ®a

the same definition as the limit except it requires  x > a . Left hand limit : lim-  f ( x ) = L . This has the  x ®a

taking x taking x large enough and positive.  x ®- ¥

 x ®a

can make  f ( x ) arbitrarily large (and positive)  by taking x taking x sufficiently close to a (on either side of a of a) without letting  x = a . There is a similar definition for  lim  f ( x ) = -¥  x ®a

except we make  f ( x ) arbitrarily large and

same definition as the limit except it requires negative.  x < a . Relationship between the limit and one-sided limits lim  f ( x ) = L Þ lim+  f ( x ) = lim- f ( x ) = L lim+  f ( x ) = lim- f ( x ) = L Þ lim  f ( x ) = L  x ®a

 x ® a

x ®a

 x ®a

 x ®a

x ®a

lim  f ( x ) ¹ lim- f ( x ) Þ lim  f ( x ) Does Not Exist

 x ®a +

x ®a

 x ®a

Properties Assume lim  f ( x ) and lim g ( x ) both exist and c is any number then,  x ®a

 x ®a

1. lim éëcf ( x ) ùû = c lim f ( x )  x ®a x ®a 2. lim éë f ( x ) ± g ( x ) ùû = lim f ( x ) ± lim g ( x )  x ®a

x ®a

x ®a

3. lim éë f ( x ) g ( x ) ùû = lim f ( x ) lim g ( x )  x ®a

x ®a

x ®a

 f ( x ) é  f ( x ) ù  xlim ®a 4. lim ê provided lim g ( x ) ¹ 0 ú=  x ®a  g x  x ®a lim g x ( ) ( ) ë û  x®a 5. lim éë f ( x )ùû = élim f ( x ) ù  x ®a ë x ®a û n

n

6. lim é n  f ( x ) ù = n lim f ( x )  x ®a

ë

û

x ®a

Basic Limit Evaluations at ± ¥

 Note : sgn ( a ) = 1 if  a > 0 and sgn ( a ) = -1 if  a < 0 . 1. 2.

lim e x = ¥

 x ®¥

&

lim lim ln ( x ) = ¥

lim e x = 0

 x ®- ¥

&

 x ®¥

3. If  r  > 0 then lim

b

 x ®¥  x r 

lim+ ln ( x ) = - ¥

 x ®0

=0

4. If  r  > 0 and is real for negative x negative x b then li lim r  = 0  x ®- ¥  x r 

5. n even : lim  x n = ¥  x ®± ¥

6. n odd : lim x n = ¥ & lim  x n = -¥  x ®¥

 x ®- ¥

7. n even : lim a x + L + b x + c = sgn ( a ) ¥ n

 x ®± ¥

8. n odd : lim a x n + L + b x + c = sgn ( a ) ¥  x ®¥

9. n odd : lim a x n + L + c x + d = - sgn ( a ) ¥

Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.

 x ®-¥

© 2005 Paul Dawkins

Calculus Cheat Sheet

Evaluation Techniques Continuous Functions L’Hospital’s Rule  f ( x ) 0  f ( x ) ± ¥ If  f ( x ) is continuous at a then lim  f ( x ) = f ( a )  x ® a If  lim then, = or  lim =  x ® a  g ( x )  x ® a  g ( x ) 0 ±¥ Continuous Functions and Composition  f ( x ) is continuous at b and lim g ( x ) = b then  x ® a

(

)

x ®a

Factor and Cancel

lim

 x ® 2

 x 2 + 4 x - 12  x 2 - 2 x

( x - 2 ) ( x + 6 ) = lim x ®2 x ( x - 2) = lim

 x + 6

 f ( x )

 x ®a  g

lim

 x ®± ¥

= =4  x 2 Rationalize Numerator/Denominator lim

 x 2 - 81

= lim

=

(18) ( 6 )

 x ® - ¥

=-

)

(

x

)

1 108

Combine Rational Expressions

1ö 1 æ x - ( x + h ) ö l i m = ç ÷ ç ÷ h ®0 h è x + h x ø h ®0 h çè x ( x + h ) ÷ø

lim

g¢ ( x)

a is a number, ¥ or  -¥

 p ( x ) q ( x)

factor largest power of  x  x in q ( x ) out

3 x 2 - 4 5 x - 2 x 2

= lim

x ®- ¥

(

 x 2 3 -

4

 x2

) = lim 3 x®- ¥ 5

x 2 ( x - 2 ) 5

 x

4

x2

-2

=-

Piecewise Function

x ®9

(

-1

lim

3- x 3+ x

x 2 - 81 3 + x 9 - x -1 = lim 2 = lim  x ®9 ( x - 81) 3 + x x®9 ( x + 9) 3 +

 x ®9

x ®a

f ¢( x)

of both  p ( x ) and q ( x ) then compute limit.

8

 x ® 2

3 -  x

( x)

= lim

Polynomials at Infinity  p ( x ) and q ( x ) are polynomials. To compute

lim  f ( g ( x ) ) = f lim g ( x ) = f ( b )

 x ®a

lim

1æ 1

1 æ -h ö 1 -1 lim = lim çç = = ÷ ÷ h®0 x ( x + h ) h ®0 h x2 è x ( x + h) ø

ì x 2 + 5 if x < -2 lim  g ( x ) where  g ( x ) = í  x ®- 2 î1 - 3 x if x ³ -2 Compute two one sided limits, lim- g ( x ) = lim- x2 + 5 = 9  x ® -2

x ® -2

lim  g ( x ) = lim+ 1 - 3 x = 7

 x ® -2+

x ® -2

One sided limits are different so lim  g ( x )  x ®-2

doesn’t exist. If the the two one sided limits had  been equal then lim  g ( x ) would have existed  x ®-2

and had the same value.

Some Continuous Functions Partial list of continuous functions and the values of  x  x for which they are continuous. 1. Polynomials for all x all x.. 7. cos ( ) and sin ( x ) for all x all x.. 2. Rational function, except for  x’s  x’s that give 8. tan ( ) and sec ( ) provided division by zero. 3p p p 3p   3. n  x (n odd) for all x all x..  x ¹ L , ,- , , ,L 2 2 2 2 n 4.  x (n even) for all  x ³ 0 . 9. cot ( x ) and csc ( x ) provided 5. e x for all x all x..  x ¹ L , -2p , -p , 0, p , 2p ,L   6. ln for  x > 0 . Intermediate Value Theorem Suppose that  f ( x ) is continuous on [a, [a, b] b] and let M  let M be be any number between  f ( a ) and  f ( b ) .

Then there exists a number c number c such that a < c < b and  f ( c ) = M  .

Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.

© 2005 Paul Dawkins

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