do now: find. hw: finish wksh 2.1 and a half – calculating limits using the limit laws
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DO NOW:Find
x
x
x
sinlim
0
xx
tanlim0
HW: Finish WKSH
2.1 and a half – Calculating Limits Using the Limit Laws
Example 1
Example 2(a) (b))43( 2
5lim
xxx x
xx
x 35
12 23
2lim
Direct SubstitutionIf f is a polynomial or a rational function and a
is in the domain of f, then
Functions with the Direct Substitution Property are called continuous at a
Remember: not all limits can be evaluated by direct substitution!
)()(lim afxfax
Example 3
In general,If f(x) = g(x) when x ≠ a, then
)()( limlim xgxfaxax
Example 4
Additional Properties of LimitsTHEOREM
If f(x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then
)()( limlim xgxfaxax
The Sandwich Theorem (AKA the Squeeze Theorem)
If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and
then
Lxhxfaxax
)()( limlim
Lxgax
)(lim
Example 5Show that 0
1sin2lim
xx
ax
Example 5 (Solution)
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