Limit of (loga (1 + x))/x

How to use the limit of Limit of (loga (1 + x))/x to solve the given limit with a logarithmic function: formula, 1 example, and its solution.

Formula

Formula

The limit of (loga (1 + x))/x as x → 0 is
1/(ln a).

Limit of (ln (1 + x))/x

Example

Example

Solution

First write the limit part and [log2 (1 + x)].

The inner part of [log2 (1 + x)] is (1 + x).

So write x
in the denominator.

To undo the denominator x,
write x in the numerator.

Write the denominator sin x.

So [log2 (1 + x)]/(sin x)
= [(log2 (1 + x))/x]⋅[x/(sin x)].

As x → 0,
(log2 (1 + x))/x → 1/(ln 2).

As x → 0,
(sin x)/x → 1.
So x/(sin x) → 1.

Limit of (sin x)/x

[1/(ln 2)]⋅1 = 1/(ln 2)

So 1/(ln 2) is the answer.