# Limit of (log_{a} (1 + x))/x

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

## Formula

## Examplelim_{x → 0} [log_{2} (1 + x)]/[sin x]

Solution

Solution (Detail)

First write the limit part and [log_{2} (1 + x)].

The inner part of [log_{2} (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 [log_{2} (1 + x)]/(sin x)

= [(log_{2} (1 + x))/x]⋅[x/(sin x)].

As x → 0,

(log_{2} (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.