Download 1.3 Evaluating Limits Analytically

1.3 Evaluating Limits Analytically
•Evaluate a limit using properties of limits.
• Develop and use a strategy for finding limits.
• Evaluate a limit using dividing out and
rationalizing techniques.
• Evaluate a limit using the Squeeze Theorem.
lim 3x =
How to read the notation. The limit of 3x as x
approaches 4 is______
Some basic limits
1. Lim b = b
2. Lim x = c
3. Lim xⁿ = cⁿ
Algebraically
1. Direct Substitution- always try first because it
is so easy. Plug in the “c” value into x and
simplify. This works for “well-behaved”
functions that are continuous at c.
a) Polynomial
If p is a polynomial function and c is a real number, the
lim p(x) = p(c).
b) Rational
If r is a rational function given by r(x) = p(x)/q(x) and c is a
real number such that q(c) ≠ 0, then lim r(x) = r(c) =
p(c)/q(c).
c) Radical
lim x = c , This limit is valid for all c if n is odd, and is
valid for c > 0 if n is even.
d) Composite Function
Find the limit of the inside function first then evaluate
the outside function.
e) Trigonometric Functions
Let c be a real number in the domain of the given
trigonometric function.
lim sin x = sin c
lim cos x = cos c
lim tan x = tan c
lim cot x = cot c
lim sec x = sec c
lim csc x = csc c
Examples
If you try substitution and get an undefined
value, #/0 or 0/0. Then you will have to use
another method to rewrite the function.
2. Factoring
3. Rationalizing the numerator
The Squeeze Theorem
It squeezes a function, that you can not find the
limit of using another method, between two
functions that you can find the limit of at c.
This theorem was used to find the following to
special trigonometric limits.
1) lim
2) lim