Introduction to :

Basic Property 1

If c is a real number, then

Think of this as taking the limit of the constant function f(x) = c. No matter what we plug in for x, we get c as the output. If we made a table

x	f(x)
x1	c
x2	c

every number in the f(x) column would be c. As x gets closer to a, f(x) gets closer to (or stays equal to) c.

As long as a is a real number, it doesn't even matter what a is. c is all that matters.

Using actual numbers, what is ?

We're looking at the limit as x approaches 3 of the function f(x) = x. If we make a table, we find

As x gets closer to 3, f(x) gets closer to 3 since x and f(x) are the same thing.

The picture of f(x) = x is a line:

From the picture we can see that as x gets closer to 3, f(x) also gets closer to 3.

We conclude that = 3.

Example 1

Find

\lim_x\to 5x.

Exercise 1

Find the limit.

\lim_x\to -30 π

Exercise 2

\lim_x\to 0\sqrt2

Exercise 3

\lim_x\to 10x

Exercise 4

\lim_x\to -4x

Exercise 5

\lim_x\to 10

Exercise 6

Find the following limit.

\lim_x\to2x

Exercise 7

\lim_x\to23x

Exercise 8

\lim_x\to210x

Exercise 9

\lim_x\to2x² + 5

Exercise 10

\lim_x\to23x² + 15

Exercise 11

\lim_x\to210x² + 50