Techniques for Computing Limits 2.3

Calculus 1

limx cK K

The limit of a constant IS the constant.No matter what “x” approaches

2lim ( )x

f x

1lim ( )x

f x

25lim ( )x

f x

Limit Laws

( )f x x

2lim ( )x

f x

1lim ( )x

f x

25lim ( )x

f x

limx cx c

lim ( ) ( ) lim ( ) lim ( )x c x c x c

f x g x f x g x

The limit of a sum is the sum of the limits.

Theorem 2.3 (1)

2

lim 5x

x

2 2

lim lim5x xx

2 5

7

lim ( ) ( ) lim ( ) lim ( )x c x c x c

f x g x f x g x

The limit of a difference is the difference of the limits.

Theorem 2.3 (2)

3

lim 7x

x

3 3

lim 7 limx x

x

7 3

4

lim ( ) lim ( )x c x ck f x k f x

The limit of a constant times a function is the constant times the limit of the function.

Theorem 2.3 (3)

2lim 6x

x 2

6 limxx

2

12

lim ( ) ( ) lim ( ) lim ( )x c x c x c

f x g x f x g x

The limit of a product is the product of the limits.

Theorem 2.3 (4)

3

limx

x x

3 3

lim limx xx x

3 3

9

lim ( )( )lim

( ) lim ( )x c

x cx c

f xf x

g x g x

The limit of a quotient is the quotient of the limits.

Theorem 2.3 (5)

6lim

3x

x

6

6

lim

lim3x

x

x

6

32

lim ( ) lim ( )nn

x c x cf x f x

Theorem 2.3 (6)

The limit of a “function raised to a power” is the “limit of the function” raised to the power.

4

2limx

x

4

2limxx

2

16

Theorem 2.3 (7)

lim ( ) lim ( )nn mm

x c x cf x f x

The limit of a “function raised to a fractional power” is the “limit of the function” raised to the fractional power.

23

8limx

x

23

8limxx

8

4

lim ( ) ( )x cp x p c

polynomialIf is a functionp

2

2lim 3 5 9x

x x

23( 2) 5( 2) 9

7

( )lim ( )

( )x c

p cr x

q c

rationalIf is funa ctionr( )

( )( )

p xr x

q x

If ( ) 0q c

2

22

5 6lim

2x

x x

x

2

2

(2) 5(2) 6

(2) 2

4

3

lim ( ) ( )x cf x f c

If is any of the six trig functionsf

for any in the domainc

lim cosx

x

cos 1

lim tanx

x

tan 0

2

22

6 8lim

4x

x x

x

0

Factor and simplify

2

2 4lim

2 2x

x x

x x

Other Techniques

1 15 5

0lim x

x x

55

5 5

115 5 xx

x

x

5 5xx

x

5 5xx x

0

15 5

limx x

1

25

9

3lim

9x

x

x

Sometimes when you have radicals you need to multiply by the conjugate of the numerator or denominator.

For all on ( , ),x a b

( ) ( ) ( )f x h x g x except possibly at c

a b

lim ( ) lim ( ) x c x cf x g x L

for some a c b

And

then lim ( )x ch x L

Squeeze TheoremPinching TheoremSandwich Theorem

2

0

1Find lim cos

xx

x

Notice we cannot substitute 0 for .xThere is no factoring or simplifying

There is no rationalizing.

We know that the range of the cosine function is [ 1,1]1

1 cos 1x

for all 0x

2Multiplying by .x

2

0limx

x

2

0limx

x

2

0

1lim cosxx

x

2

0

1lim cos 0xx

x

2

0lim 0x

x

2

0lim 0x

x

Piecewise2 1

( )1

xf x

x

If x < -1

If x ≥ -1

1lim ( )x

f x

1lim ( )x

f x

1

lim ( )x

f x