Trigonometric Limits

[Pages:40]Trigonometric Limits

more examples of limits

Substitution Theorem for Trigonometric Functions

laws for evaluating limits

Theorem A. For each point c in function's domain:

lim sin x = sin c,

xc

lim tan x = tan c,

xc

lim csc x = csc c,

xc

lim cos x = cos c,

xc

lim cot x = cot c,

xc

lim sec x = sec c.

xc

Theorem A. For each point c in function's domain:

lim sin x = sin c,

xc

lim tan x = tan c,

xc

lim csc x = csc c,

xc

lim cos x = cos c,

xc

lim cot x = cot c,

xc

lim sec x = sec c.

xc

Proof. Prove first that

lim sin x = 0,

x0

lim cos x = 1.

x0

Is it obvious? lim sin x = 0,

x0

lim cos x = 1.

x0

y=sin(x)

y=cos(x)

Is it obvious? lim sin x = 0,

x0

lim cos x = 1.

x0

y=sin(x)

y=cos(x)

No. The picture is not precise.

Is it obvious? lim sin x = 0,

x0

lim cos x = 1.

x0

y=sin(x)

y=cos(x)

No. The picture is not precise. Use definitions of sin(x) and cos(x).

Use The One-Sided Squeeze Theorem. If

f (x) g(x) h(x) near c and lim f (x) =

xc+

lim h(x) = Lright, then

xc+

y=h(x)

L y=g(x)

lim g(x) = Lright

xc+

y=f(x) c

Also, if lim f (x) = lim h(x) = Lleft, then

xc-

xc-

y=h(x)

y=g(x) L

lim g(x) = Lleft y=f(x)

xc-

c

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