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Different Types of Limits
Besides ordinary, two-sided limits, there are one-sided limits (left-
hand limits and right-hand limits), infinite limits and limits at infinity.
√ One-Sided Limits
Consider limx→5 x2 −4x−5.
One might think that since x2 −4x−5 → 0 as x → 5, it would follow
that lim √x2−4x−5=0.
x→5
But since x2 − 4x − 5 = (x − 5)(x + 1) < 0 when x is close to 5 but
smaller than 5, √x2 −4x−5 is undefined for some values of x very
close to 5 and the limit as x → 5 doesn’t exist.
But we would still like a way of saying √x2 − 4x − 5 is close to 0 when
x is close to 5 and x > 5, so we say the Right-Hand Limit exists, write
lim + √x2 −4x−5 = 0 and say √x2−4x−5 approaches 0 as x
x→5
approaches 5 from the right.
Sometimes we have a Left-Hand Limit but not a Right-Hand Limit.
SometimeswehavebothLeft-HandandRight-HandLimitsandthey’re
not the same. Sometimes we have both Left-Hand and Right-Hand
Limits and they’re equal, in which case the ordinary limit exists and is
the same.
Example
x2 if x < 1
f(x) = x3 if 1 < x < 2
x2 if x > 2.
lim − f(x) = lim + f(x) = 1, so the left and right hand limits
x→1 x→1
are equal and lim f(x)1.
x→1
lim − f(x) = 8 while lim + f(x) = 4, so the left and right hand
x→2 x→2
limits are different and lim f(x) doesn’t exist.
x→2
Limits at Infinity
Suppose we’re interested in estimating about how big 2x is when
x+1
xis very big. It’s easy to see that 2x = 2x = 2 ifx6=−1
x+1 x(1+ 1) 1+1
2x x x
and thus x+1 will be very close to 2 if x is very big. We write
lim 2x =2
x→∞ x+1
and say the limit of 2x is 2 as x approaches ∞.
x+1
1
2
Limits at Infinity
Similarly, 2x will be very close to 2 if x is very small and we write
x+1
lim 2x =2
x→−∞ x+1
and say the limit of 2x is 2 as x approaches −∞. Here, small does
x+1
not mean close to 0, but it means that x is a negative number with a
large magnitude (absolute value).
Calculating Limits at Infinity
Aconvenient way to find a limit of a quotient at infinity (or minus
infinity) is to factor out the largest term in the numerator and the
largest term in the denominator and cancel what one can.
5x2 −3 x2(5− 3 )
2
lim =lim x =
x→∞ 8x2 −2x+1 x→∞ x2(8− 2 + 1 )
2
x x
5− 3 5
2
lim x =
x→∞ 8− 2 + 1 8
2
x x
Example
5x−3 x(5− 3)
lim =lim x =
x→∞ 8x2 −2x+1 x→∞ x2(8− 2 + 1 )
2
x x
5−3
lim x =0
x→∞ x(8− 2 + 1 )
2
x x
Infinite Limits
If x is close to 1, it’s obvious that 1 is very big. We write
2
1 (x−1)
lim =∞
x→1 2
(x−1)
and say the limit of 1 is ∞ as x approaches 1.
2
(x−1)
Similarly, lim − 1 =−∞.
x→1 2
(x−1)
ATechnicality
Technically, a function with an infinite limit doesn’t actually have
a limit. Saying a function has an infinite limit is a way of saying it
doesn’t have a limit in a very specific way.
Calculating Infinite Limits
3
Infinite limits are inferred fairly intuitively. If one has a quotient
f(x), one may look at how big f(x) and g(x) are. For example:
g(x)
If f(x) is close to some positive number and g(x) is close to 0 and
positive, then the limit will be ∞.
If f(x) is close to some positive number and g(x) is close to 0 and
negative, then the limit will be −∞.
If f(x) is close to some negative number and g(x) is close to 0 and
positive, then the limit will be −∞.
If f(x) is close to some negative number and g(x) is close to 0 and
negative, then the limit will be ∞.
Variations of Limits
One can also have one-sided infinite limits, or infinite limits at infin-
ity.
1
lim + =∞
x→1 x−1
1
lim − =−∞
x→1 x−1
Asymptotes
If lim f(x) = L then y = L is a horizontal asymptote.
x→∞
If lim f(x) = L then y = L is a horizontal asymptote.
x→−∞
If lim + f(x) = ±∞ then x = c is a vertical asymptote.
x→c
If lim − f(x) = ±∞ then x = c is a vertical asymptote.
x→c
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