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 limx→5√x2−4x−5 = 0.
But since x2−4x−5=(x−5)(x+ 1) <0 when xis close to 5 but
smaller than 5, √x2−4x−5 is undefined for some values of xvery
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
xis close to 5 and x > 5, so we say the Right-Hand Limit exists, write
limx→5+√x2−4x−5 = 0 and say √x2−4x−5approaches 0as x
approaches 5from the right.
Sometimes we have a Left-Hand Limit but not a Right-Hand Limit.
Sometimes we have both Left-Hand and Right-Hand Limits and they’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
f(x) =
x2if x < 1
x3if 1 < x < 2
x2if x > 2.
limx→1−f(x) = limx→1+f(x) = 1, so the left and right hand limits
are equal and limx→1f(x)1.
limx→2−f(x) = 8 while limx→2+f(x) = 4, so the left and right hand
limits are different and limx→2f(x) doesn’t exist.
Limits at Infinity
Suppose we’re interested in estimating about how big 2x
x+ 1 is when
xis very big. It’s easy to see that 2x
x+ 1 =2x
x(1 + 1
x)=2
1 + 1
x
if x6=−1
and thus 2x
x+ 1 will be very close to 2 if xis very big. We write
limx→∞
2x
x+ 1 = 2
and say the limit of 2x
x+ 1 is 2as xapproaches ∞.
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