limit x->0 ln(1+x)/x — step-by-step solution

Compute the limit of

\frac{\ln (1 + x)}{x}

x \to 0

The limit of

1 + x

1

when

x \to 0

The limit of

\ln (1 + x)

0

when

x \to 0

For

\frac{\ln (1 + x)}{x}

, we encounter the indeterminate form 0/0 as

x

tends to

0

We can use L'Hôpital's rule (limif of a/b = limit of a'/b') by computing the derivatives of the numerator and the denominator:

Let's perform the differentiation:

{(\ln (1 + x))}^{'}

Apply the differentiation rule:

{(\ln (f))}^{'} = {(f)}^{'} \cdot \frac{1}{f}

with f =

1 + x

{(1 + x)}^{'} \cdot \frac{1}{1 + x}

\frac{0 + 1}{1 + x}

\frac{1}{1 + x}

Let's differentiate:

{(x)}^{'}

1

With the derivatives of the numerator and the denominator, we obtain a new fraction whose limit can be determined:

The limit of

\frac{1}{1 + x}

1

when

x \to 0

So the limit of

\frac{\ln (1 + x)}{x}

1

when

x \to 0

▷Answer: $1$