Calculus
Limits
Limit of a function……………………………………………..……………1 One-Sided Limits……………………………………………….…..………..1 Infinite limits…………………………………………………………………2 Vertical Asymptotes.........................................................................................3 Calculating Limits Using the Limit Laws…………… …………………….5 The Squeeze Theorem ………………………………… ………………….6 The Precise Definition of a Limit…………………..…….……………….....7 Continuity…………………………………………………………………….8 Intermediate Value Theorem……….……………………………………….9 References……………………………………………………………………..…9
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Calculus
Limits
Limit of a function Lim f ( x) = L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close
x >a
to a but not equal to a.
Example: Let f(x) =
t2 +9 3 . Discuss the behavior of the values of f(x) when x is close to 0. t2
t
Solution: Make a table to see the behavior…
± 1.0 ± 0.5 ± 0.1 ± 0.05
t2 +9 3 lim = 0.16666 t →0 t2
t2 +9 3 t2
0.16228 0.16553 0.16662 0.16666
As t approaches 0, the values of the function seem to approach 0.16666…
One-Sided Limits Definition: We write Lim f ( x ) = L and say the left-hand limit of f(x) as x approaches a is equal to L if
x→a
we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and less than a i.e x approaches a from the left.
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Calculus
Limits
Definition: We write Lim f ( x ) = L and say the right-hand limit of f(x) as x approaches a is equal to L +
x→a
if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and greater than a i.e x approaches a from the right
Lim f ( x) = L if and only if Lim f ( x) = L and Lim f ( x) = L +
x→a x→a x→a
Infinite 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 by taking x sufficiently close

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