x→a
to a, but not equal to a.
2
Calculus
Limits
Definition: Let f be a function defined on both sides of a. Then Lim f (x) = ∞ means that the values of f(x) can be made arbitrarily large negative by taking x
x→a
sufficiently close to a, but not equal to a.
Vertical Asymptotes
Definition: The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true:
Lim f (x ) = ∞
x→a
Lim f (x ) = ∞
x→a
Lim f (x ) = ∞ +
x→a x→a
Lim f (x) = ∞
x→a
Lim f (x ) = ∞
x→a
Lim f (x) = ∞ +
3
Calculus
Limits
Example: Find the vertical asymptotes of f(x) = tanx Solution: Because tan x =
sin x cos x
x → (π ) and cos x → 0 as x → (π ) + , whereas sin x is positive when x is near π , we have 2 2 2 lim tan x = ∞ and lim tan x = ∞ . This shows that the line x = π is a vertical asymptote. 2 x →( π x →( π 2)+
2)
There are potential vertical asymptotes where cosx = 0. In fact, since
cos x → 0 + as
Similar reasoning shows that the lines
x = (2n + 1) π , (odd multiples of π ),where n is an integer, 2 2
are all vertical asymptotes of f(x) = tanx.
4
Calculus
Limits
Calculating Limits Using the Limit Laws
Suppose that c is a constant and the limits Lim f (x ) and Lim g (x ) exist. Then
x→ a x→ a
1. Lim[ f ( x ) + g ( x )] = lim f ( x ) + lim g ( x )
x→ a x→ a x→a x→a

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