# Section 2.1: Introduction to Limits

## Definition of a Limit

We write:limx → c f(x) = Lorf(x) → Lasx → c

if the functional value f(x) is close to the single real number L whenever x is close, but not equal, to c (on either side of c.)

Note that:

• The existence of a limit at c has no relation to the value of the function at c.
• c does not even need to be in the domain of f.
• The function must be defined on both sides of c.

Drag the slider to see the value of the function f(x) = x + 1 as x is close to 1.

x = 1.5, f(x) = 2.5

## One-Sided Limits

We write:limx → c- f(x) = K

and call K the limit from the left or the left-hand limit if f(x) is close to K whenever x is close to, but to the left of, c on the real number line.

We write:limx → c+ f(x) = L

and call L the limit from the right or the right-hand limit if f(x) is close to L whenever x is close to, but to the right of, c on the real number line.

### On the Existence of a Limit

For a (two-sided) limit to exist, the limit from the left and the limit from the right must exist and be equal. That is,

limx → c f(x) = Lif and only iflimx → c- f(x) = limx → c+ f(x) = L

Drag the slider to see how the value of the function f(x) changes with x.

x = -0.5, f(x) = 1

## Properties of Limits

Let f and g be two functions, and assume that:

limx → c f(x) = Landlimx → c g(x) = M

where L and M are real numbers (both limits exist). Then:

 limx → ck = k limx → cx = x limx → c[f(x) + g(x)] = limx → cf(x) + limx → cg(x) = L + M limx → c[f(x) - g(x)] = limx → cf(x) - limx → cg(x) = L - M limx → ck f(x) = klimx → cf(x) = kL limx → c[f(x) ⋅ g(x)] = [limx → cf(x)] [limx → cg(x)] = LM limx → c[f(x)/g(x)] = [limx → cf(x)]/[limx → cg(x)] = L/M if M ≠ 0 limx → cn√ f(x) = n√ limx → c f(x) = n√ L, L > 0 for n even

## Limits of Polynomial and Rational Functions

• limx → cf(x) = f(c) for f any polynomial function.
• limx → cr(x) = r(c) for r any rational function with a nonzero denominator at x = c.

## Indeterminate Form

• If limx → cf(x) = 0 and limx → cg(x) = 0,

then limx → cf(x)/g(x) is said to be indeterminate, or, more specifically, a 0/0 indeterminate form.

• The limit of an indeterminate form may or may not exist.

## Limit of a Quotient

If limx → cf(x) = L, L ≠ 0 , and limx → cg(x) = 0

then limx → cf(x)/g(x) does not exist.