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Material Type: Notes; Professor: Lamb; Class: MATH 1910: If high school precalculus and ACT math of at least 26 contact 694-6450.; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 1989;
Typology: Study notes
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Section 2.3 Calculating Limits Using the Limit Laws
Look at the limit laws beginning on page 111 of the textbook. An important note is that the limit laws can only be applied to functions whose limits EXIST.
Ex. Given that lim ( ) x a f x → = 2 lim ( ) x a g x → = − 4 lim ( ) x a h x →
evaluate the following limits, if they exist.
a) lim[ ( ) ( )] x a f x h x → 3 − b) lim[ ( )] x a g x →
(^2) c) lim (^ ) ( ) x a (^ )
g x → h x^ + f^ x
d) lim (^) ( ) x → a f^ x
(^1) e) lim ( ) ( ) x a h x g x →
Ex. Evaluate the following limits, if they exist, without the aid of a calculator:
a) lim x
x →− x x
b) lim x x x →−
1
There are times when you have to do some algebraic maneuvering such as factoring both sides of a rational expression, rationalizing the NUMERATOR of a rational expression before you can evaluate a limit.
Ex. Evaluate the following limits, if they exist, without the aid of a calculator:
a) lim x
x x →− x
b) lim x
x → x
2 3
c) lim h
h → h
0
lim (^ )(^ ) x
x x (^3) x
= lim (^ )(^ ) x ( )( )
x x → x x x
= lim h
h h
h → h
cancel out the ( x + 3 ) = lim (^ ) x ( )
x → x x
rationalize the numerator
= − →− lim ( ) x x 3
= lim h ( )
h → h h
= lim h ( )
h → (^0) h ⋅ 5 + h + 5 = lim h → (^) + h +
0
Ex. Evaluate the one sided limit lim x → +^ x x
(^1 1) using your knowledge of the absolute value function.
Section 2.3 Calculating Limits Using the Limit Laws
Ex. For the function f x
e x x x x x
x ( )
. Which of the following limits exist:
a) lim ( ) x f x → 0 b) lim ( ) x f x → 1 c) lim ( ) x f x → 2 HINT : Try not to graph the function, just use the y-values on the different intervals and your knowledge of the functions involved.
THE SQUEEZING THEOREM : If f ( ) x ≤ g x ( ) ≤ h x ( )when x is near a (except possible at a) and lim ( ) lim ( ) x a x a f x h x L → → = = then lim ( ) x a g x L →
Ex. Use the Squeeze Theorem to show that lim sin x x x →
0
First, notice we can’t use the limit laws and split the limit up as lim lim sin x x x x → →
0 0
x → 0 x
How can we now apply the squeeze theorem?