Section 11.2 One-Sided Limits and Limits Involving Infinity:
1. Graph the function f (x) =
|x|
and find lim .
x→0
x
2. One-Sided Limits: Let f be a function, and let a, K and M be real numbers. Assume that
f (x) is defined for all x near a, with x > a.
Suppose that as x takes values very close (but not equal) to a (with x > a), the corresponding
values of f (x) are very close (and possibly equal) to K.
And that the values of f (x) can be made as close as you want to K for all values of x (with x > a)
that are close enough to a.
Then the number K is the limit of f (x) as x approaches a from the right, which is written
lim f (x) = K.
x→a+
The number M is the limit of f (x) as x approaches a from the left, which is written
lim f (x) = M
x→a−
is defined in a similar fashion, with the obvious changes.
3. Two-Sided Limits Let f be a function, and let a and L be real numbers. Then
lim f (x) = L exactly when lim− f (x) = L and lim+ f (x) = L
x→a
x→a
x→a
(Note, all properties/theorems about limits in Section 11.1 are valid for left- and right-handed limits.)
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4. Use the following graph to answer the below questions:
(a)
(b)
lim f (x)
(f) lim− f (x)
x→−3−
x→2
lim f (x)
(g) lim+ f (x)
x→−3+
x→2
(c) lim f (x)
(h) lim f (x)
(d) f (−3)
(i) f (2)
(e) For which value(s) of x is
f (x) = −3?
(j) For which value(s) of x is
f (x) = 2?
x→2
x→−3
5. Find
lim+
x→2
√
4 − x2 and lim−
x→2
2
√
4 − x2
6. Find
lim+
√
x − 3 + x2 + 1
x→3
The cost, c(x) of a NYC taxicab fare during
nonpeak hours (assuming no standing time)
where x is the distance of the trip (in miles)
is $3 upon entry plus $0.40 per fifth of a mile.
Use the accompanying figure to answer the
following questions.
7.
(a) c(0)
(b) c(1.2)
(c) limx→1.2− c(x)
(d) limx→1.2+ c(x)
(e) limx→1.2 c(x)
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8. The below graph shows different infinite limits:
9. Limits at Infinity Let f be a function that is defined for all large positive values of x and let L
be a real number. Suppose that
as x takes larger and larger positive values, increasing without bound, the corresponding values
of f (x) are very close (and possibly equal to) L. and that the values of f (x) can be made arbitrarily
close (as close as you want) to L by taking large enough values of x.
Then we say that the limit of f (x) as x approaches infinity is L, which is written
lim f (x) = L.
x→∞
It can be similarly defined for small negative values of x and Then we say that the limit of f (x)
as x approaches negative infinity is M , which is written
lim f (x) = M.
x→−∞
1
1
and lim . Write a table of values making x get larger and larger, or smaller
x→∞ x
x→−∞ x
10. Consider: lim
and smaller.
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11. The following two hold true:
1
=0
x→∞ xn
lim
and
1
= 0.
x→−∞ xn
lim
Note: ‘Show all algebraic work’ means you MUST use the above material.
12. Find the following limits
8x2 + x + 6
x→∞
3x2 − 1
(a) (Show all algebraic work) lim
8x + 6
x→∞ 3x2 − 1
(b) lim
8x2 + 6
x→∞ 3x − 1
(c) lim
x3 + 8x2 + 6
x→−∞ x2 − 3x − 1
(d) lim
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