Transcript Limits: An Algebraic Perspective

Indeterminate Forms
Recall that, in some cases, attempting to
evaluate a limit using substitution yields an
indeterminate form, such as 0/0.
Usually, we can use factoring if the
function is a rational function…
… or we can multiply by the conjugate if
the function has radicals, etc.
Indeterminate Forms
But what if the function and the limit
involve trigonometry, such as:
sin x
lim
x
x→0
Substitution yields an indeterminate form,
but can we rewrite our expression to “fix”
the 0/0 issue?
Indeterminate Forms
 Graphically, this limit
clearly exists (and
equals 1).
 We cannot use any of
our previously studied
methods to “fix” the
function.
 Let’s look at other
functions with similar
behavior around x=0.
The Squeeze Theorem
 In the neighborhood
of x=1, the graph of
f(x)=(sin x)/x is
“squeezed” by the
graphs of g(x)=cos x
and h(x)=1.
 Because the limits of
g(x) and h(x) both
equal 1 as x
approaches 0, so
must that limit on f(x).
The Squeeze Theorem
Let f, g, and h be defined on an interval
containing c (except possibly at c itself).
For every x other than c in that interval,
g(x)<f(x)<h(x).
If lim g(x) = lim h(x) = L , then
x→c
lim f(x) = L
x→c
x→c
Trigonometric Limits
The squeeze theorem is useful for finding
trigonometric limits, including:
sin x
lim
=1
x
x→0
1 – cos x
lim
=0
x
x→0
But it’s not always necessary…
Trigonometric Limits
Let’s look at strategies for evaluating:
tan x
lim
x
x→0
sin 4x
lim
x
x→0
lim xcos x
x→0
Techniques for Evaluating Limits
Substitution
Simplification using factoring
Multiplication by the conjugate
Analysis of infinite limits
Squeeze theorem
Change of variables