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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