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Formal vs Informal Definition of a Limit
Definition 1 (Informal definition). The limit of a function f (x), as x ap- proaches a ∈ R, is L ∈ R, and we write
lim x→a f (x) = L
if the values of f (x) can be made arbitrarily close to L by choosing the values of x close enough to a.
Definition 2 (Formal definition). We write
lim x→a f (x) = L
if for any ε > 0 there exists δ > 0 such that
|x − a| < δ implies |f (x) − L| < ε.
Computing limits by definition To prove that lim x→a f (x) = L
consider a game in which your opponent makes a move by giving you a number ε > 0 and you respond by producing a number δ > 0 (which depends on ε) such that |x − a| < δ implies |f (x) − L| < ε
Example 1. Prove that lim x→ 1
Solution. We have f (x) = 1 for every x. Suppose ε > 0 is given and we have to find δ > 0 such that |x − 1 | < δ implies | 1 − 1 | < ε. In this case any positive number δ works since | 1 − 1 | = 0 < ε. So, choose δ = 1 - this is our response to any given ε.
Computing limits by definition
Example 2. Prove that lim x→ 1
x = 1
Example 3. Prove that lim x→ 0 x^3 = 0
One-sided limits Similarly, one can defined one-sided limits as follows
Definition 3. We write lim x→a−^
f (x) = L
if the values of f (x) can be made arbitrarily close to L by choosing the values of x close enough to a and less than a. We write lim x→a+^
f (x) = L
if the values of f (x) can be made arbitrarily close to L by choosing the values of x close enough to a and greater than a.
One-sided limits
Fact 1. lim x→a f (x) = L
if and only if
lim x→a−^
f (x) = L and lim x→a+^
f (x) = L.
Example 4.
lim x→ 0
|x| x
= undefined
Basic techniques
Fact 2. The limit of a sum, difference, or product is equal to the sum, difference, or product of the limits provided all the limits involved exist. The limit of a quotient is the quotient of the limits provided that the limit of the denominator is not 0.
Example 5 (Positive example).
lim x→ 0
x^3 + 2x^2 − 1 1 − 3 x
Example 6 (Negative example).
lim x→ 2
x^2 − 4 x − 2
Basic techniques
Fact 3. If f (x) = g(x) for every x near a (but not equal to a) then
lim x→a f (x) = lim x→a g(x)
provided both exist.
Example 11.
lim x→ 1 −
x − 1
= −∞ lim x→ 1 +
x − 1
Example 12. lim x→ 0 +^
ln x = −∞
Example 13. lim x→ π 2 −
tan x = ∞
Limits at infinity
Definition 6. We write lim x→∞ f (x) = L
if the values of f (x) can be made arbitrarily close to L by choosing the values of x large enough.
We write lim x→−∞ f (x) = L
if the values of f (x) can be made arbitrarily close to L by choosing the values of x large enough negative.
In either case we say that the horizontal line y = L is a horizontal asymptote for f (x).
Limits at infinity
Fact 6. If r > 0 then
lim x→∞
xr^
= 0, lim x→−∞
xr^
Example 14.
lim x→∞
x^2 − 1 x^2 + 1
Limits at infinity
Example 15.
lim x→∞
x + 2 √ 9 x^2 + 1
Example 16.
lim x→−∞
x + 2 √ 9 x^2 + 1
Limits at infinity
Example 17. lim x→−∞ ex^ = 0
Example 18.
lim x→∞
x^2 ex^
Example 19. lim x→∞ sin x = undefined
Continuity at a point
Definition 7. A function f (x) is continuous at a ∈ R if
lim x→a f (x) = f (a)
Remark 1. Continuity of f (x) at a implies that f (a) is defined (a ∈ Dom(f )), limx→a f (x) exists, and the equality from the definition holds.
Definition 8. If f (x) is not continuous at a then it is called discontinuous at a. There are three types of discontinuity.
Types of discontinuity
if a /∈ Dom(f ) but it is possible to extend f to a new function f such that
f (x) =
f (x), if x ∈ Dom(f ), limx→a f (x), if x = a.
Example 20. Check continuity of the function
f (x) =
x^2 − 1 x − 1
at the point x = 1.
Types of discontinuity
one of the limits limx→a− f (x), limx→a+ f (x) is equal to ±∞.
Example 24. Find the values of the parameter c which make the function continuous everywhere
f (x) =
cx^2 + 2x if x < 2 , x^3 − cx if x > 2 ,
Continuous functions
Fact 9 (Composition). If f (x) is continuous at b and limx→a g(x) = b, then
lim x→a f (g(x)) = f
lim x→a g(x)
Example 25.
lim x→ 2 tan−^1
x^2 − 4 3 x^2 − 6 x
= tan−^1
Intermediate Value Theorem
Theorem 1. If f (x) is continuous on the closed interval [a, b] and N ∈ R is between f (a) and f (b), where f (a) ̸= f (b), then there exists c ∈ [a, b] such that f (c) = N.
Example 26. Show that there is a root of the equation x^4 + x − 3 = 0 on the interval (1, 2).
