Limit of a Function

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

Formula

Formula

See the graph of y = f(x).

As x goes to a,
the graph goes to α.

Then you can write the like below:
limx → 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

Example 1

Example

Solution

To find the limit of [√x2 + 5 + 3x]
as x → 2,
put 2 into [√x2 + 5 + 3x].

Solve the expression.

Then (given) = 9.

So 9 is the answer.

Example 2

Example

Solution

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.