Indeterminate Form (Part 2)

Indeterminate Form (Part 2)

How to solve the limits of functions in indeterminate form [0/0, 0⋅∞]: examples and their solutions.

Example 1: 0/0

Find the limit of the given expression. The limit of (x^2 + x - 2)/(x - 1) as x goes to 1.

If you put 1 into the x-s,
both of the numerator and the denominator become 0,
which yields 0/0 form.

To solve this,
find the factor that makes 0, (x - 1)
and cancel (x - 1).

To find (x - 1),
factor the numerator.

Factoring a quadratic trinomial

Cancel x - 1,
which makes 0/0 form.

Put 1 into the x.

Then (limit) = 3.

Example 2: 0/0

Find the limit of the given expression. The limit of [sqrt(x + 8) - 3]/(x - 1) as x goes to 1.

To solve 0/0 form with a square root,
multiply the conjugate of the square root part
(√x + 7 + 3)
to both of the numerator and the denominator.

It's like rationalizing the numerator
to find (x - 2) factor,
which makes 0/0 form.

Rationalizing a denominator

Cancel x - 2,
which makes 0/0 form.

Put 2 into the x.

Example 3: 0 × ∞

Find the limit of the given expression. The limit of (1/x)*[6/(x + 3) - 2] as x goes to 0.

If you put 1 into the x-s,
1/x becomes ∞
and 6/(x + 3) - 2 becomes, 2 - 2, 0,
which yields 0 × ∞ form.

To solve this,
change 6/(x + 3) - 2
to find the factor (x). (blue)

Adding and subtracting rational expressions

Cancel x,
which makes 0 × ∞ form.

Put 0 into the x.