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

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

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

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.