# Limit of a Function

How to find the limit of the given function: definition, 2 examples, and their solutions.

## Definition

See the graph of y = f(x).

As x goes to a,

the graph goes to α.

Then you can write the like below:

lim_{x → a} f(x) = α.

The left side is read as

[the limit of f(x) as x goes to a].

The limit of f(x) means

where the graph of y = f(x) is going.

It doesn't mean the function value f(a).

So the limit of f(x) and f(a)

can or cannot be equal.

Continuous

## Examplelim_{x → 2} (√x^{2} + 5 + 3x)

To find the limit of [√x^{2} + 5 + 3x]

as x → 2,

put 2 into [√x^{2} + 5 + 3x].

Solve the expression.

2^{2} = 4

3⋅2 = 6

4 + 5 = 9

√9

= √3^{2}

= 3

Square Root

3 + 6 = 9

So 9 is the answer.

## Examplelim_{x → 1} f(x) and f(1)

Recall that

the limit of f(x) means

where the graph of y = f(x) is going,

not the function value.

So, to find the limit of f(x) as x → 1,

you should choose the case when x ≠ 1:

x + 2.

So the limit of f(x) is

the limit of x + 2.

To find the limit of x + 2

as x → 1,

put 1 into x + 2.

1 + 2 = 3

So 3 is the answer.