Download Sec. 1.3: Evaluating Limits Analytically

Sec. 1.3: Evaluating Limits Analytically
• The limit of f(x) as x approaches c does not
depend on the value of f at c.
i.e. The limit of f(x) as x approaches c may not
be f(c).
• Although, for those that are, we could have
used direct substitution to evaluate the limit.
Limits Using Direct Substitution
If b and c are real numbers and n is a positive
integer, then
1. lim b  b 2. lim x  c 3. lim x  c
n
xc
xc
x c
n
More Limits Using Direct Substitution
If p is a polynomial function, then
lim p( x)  p(c)
x c
If r is a rational function r(x) = p(x)/q(x), then
p (c )
lim r ( x)  r (c) 
x c
q (c )
q (c )  0
More Limits Using Direct Substitution
For radical functions, if n is positive, then the
following limit is valid for all c if n is odd, and
all c > 0 if n is even.
lim x  c
n
x c
n
More Limits Using Direct Substitution
For trigonometric functions, if c is in the
domain of the function, then
1. lim sin x  sin c 2. lim cos x  cos c
x c
x c
3. lim tan x  tan c 4. lim cot x  cot c
x c
x c
5. lim sec x  sec c 6. lim csc x  csc c
x c
x c
Properties of Limits (Rules)
Let b and c be real numbers, let n be a positive integer, and let
f and g be functions with the following limits.
lim f ( x)  L and
x c
1. Scalar multiple:
lim g ( x)  K
x c
lim b  f ( x)  b  L
x c
2. Sum or difference: lim  f ( x)  g ( x)   L  K
x c
3. Product:
lim  f ( x)  g ( x)   L  K
x c
4. Quotient:
 f ( x)  L
lim 
 , K 0

x c g ( x )

 K
5. Power:
lim  f ( x)   Ln
n
x c
More Properties of Limits
The limit of a composite function:
If f and g are functions such that
lim g ( x)  L and
x c
then

lim f ( x)  f ( L)
xL

lim f ( g ( x))  f lim g ( x)  f ( L)
x c
x c
What if direct substitution won’t
work?
Try the following:
If direct substitution gives 0/0, then
1. Factor and use the dividing out strategy.
2. Rationalize and use the dividing out strategy.
3. Simplify and use the dividing out strategy.
Still won’t work? Fall back on using a graph
or a table.