2.2 The limit of a function EXAMPLE Refer to this graph of f(x) Notice that when x is close to 1, f(x) is close to 2. That is, if x is taken from small interval containing 1, f(x) will be in a small interval containing 2. As the x-­‐interval containing 1 gets smaller, the y-­‐interval containing 2 will get smaller. DEFINITION (intuitive) Suppose that for a function f, the value f(x) can be made arbitrarily close to L by making x arbitrarily close (but not equal to) the number a. Then we say that “the limit of f(x), as x approaches a, equals L.” lim f (x) = L x→a
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EXAMPLE Use a calculator or spreadsheet to investigate x−2
lim 2
x→2 x − 2x
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We will evaluate the function x−2
f (x) = 2
x − 2x
for a succession of x values that are getting closer and closer to x = 2 from the left (that is, values of x that are less than 2), then repeat the process for a succession of x values that are getting closer and closer to x = 2 from the right (values of x that are greater than 2). x 1.5 1.9 1.99 1.999 1.9999 x 2.5 2.1 2.01 2.001 2.0001 It appears that, as x gets closer and closer to 2 from the left, f(x) gets closer and closer to 0.5. We say that the limit of f(x), as x approaches 2 from the left, equals 0.5. lim f (x) = 0.5 x→2 −
This value is also called “the left-­hand limit as x approaches 2.” €
It also appears that, as x gets closer and closer to 2 from the right, f(x) gets closer and closer to 0.5. We say that the limit of f(x), as x approaches 2 from the right, equals 0.5. lim f (x) = 0.5 x→2+
This is also called “the right-­hand limit as x approaches 2.” €
Because the left-­‐hand limit is the same as the right-­‐hand limit, we say that the limit of f(x) as x approaches 2 exists and is equal to 0.5. lim f (x) = 0.5 x→2
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If the left-­‐hand limit were different from the right-­‐hand limit, we would say that the limit of f(x), as approaches 2, does not exist. FACT lim f (x) = L if and only if lim− f (x) = L and lim+ f (x) = L x→a
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x→a
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x→a
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Refer to the graph below. Evaluate lim f (x) lim f (x) x→2 −
x→2+
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lim f (x) f(2) x→2
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Refer to the graph below. Evaluate lim f (x) lim f (x) lim f (x) x→1−
x→1+
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f(1) x→1
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INFINITE LIMITS Use a calculator or spreadsheet to investigate 1
lim
2
x→1 ( x −1)
For the left-­‐hand limit will use x values that are less than 1 but getting successively closer to 1. €
x .9 .99 .999 €
.9999 .99999 f (x) =
1
( x −1)
2
For the right-­‐hand limit we will use x values that are greater than 1 but getting successively closer to 1. x 1.1 1.01 1.001 €
1.0001 1.00001 f (x) =
1
( x −1)
2
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It appears that as x approaches 0 from the left, f(x) increases without bound. We say that the limit of f(x), as x approaches 0 from the left, is infinity. 1
= ∞ lim−
2
x→1 ( x −1)
Likewise, it appears that as x approaches 0 from the right, f(x) increases without bound. We say that the limit of f(x), as x approaches 0 from the right, is infinity. 1
= ∞ lim+
2
x→1 ( x −1)
Since the left-­‐hand limit is the same as the right-­‐hand limit €
lim
x→1
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1
( x −1)
2
= ∞ DEFINITIONS Let f be a function defined on both sides of a, except possibly at a itself. Then lim f (x) = ∞ x→a
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means that values of f(x) can be made arbitrarily large by taking values sufficiently close to a; lim f (x) = −∞ x→a
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means that values of f(x) can be made arbitrarily large negative by taking values sufficiently close to a. Left-­‐hand and right-­‐hand infinite limits are defined similarly. Infinite limits are associated with vertical asymptotes. In the previous example, the function f (x) =
1
( x −1)
2
has a vertical asymptote when x = 1. €
DEFINITION The line x = a is a vertical asymptote for the curve y = f(x) if at least one of the following is true: lim f (x) = ∞ lim f (x) = ∞ lim f (x) = ∞ x→a−
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x→a+
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lim− f (x) = −∞ x→a
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x→a
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lim+ f (x) = −∞ x→a
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lim f (x) = −∞ x→a
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Infinite limits and their associated vertical asymptotes typically occur when a function is a quotient and a is a value that makes the denominator zero while the numerator is nonzero. Examples: The line x = –2 is a vertical asymptote for the function x
f (x) =
x+2
As x approaches –2, the left-­‐hand limit is infinity and the right-­‐hand limit is negative infinity. The lines x = 0, x = π, x=nπ, x = –nπ are the vertical cos x
asymptotes for the function f (x) = cot x =
sin x
In each case the left-­‐hand limit is negative infinity and the right-­‐hand limit is infinity.
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