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:


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) = nlimx → c f(x) = n√ L, L > 0 for n even

Limits of Polynomial and Rational Functions

Indeterminate Form

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.