DO NOW: Find. HW: Finish WKSH 2.1 and a half – Calculating Limits Using the Limit Laws.

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)