# How do you find the limit lnx/x as x->oo?

Nov 5, 2017

${\lim}_{x \to \infty} \ln \frac{x}{x} = 0$

#### Explanation:

If we evaluate the limit of the numerator and denominator separately we'll find that:

$\cdot$ As $\ln \left(x\right)$ goes to $\infty$ as $x$ goes to $\infty$: $\ln \left(\infty\right) = \infty$

$\cdot$ $x$ goes to $\infty$

Therefore we have a ratio of two infinities $\frac{\infty}{\infty}$ meaning that we will have to apply L'Hospital's Rule.

${\lim}_{x \to \infty} \ln \frac{x}{x} = {\lim}_{x \to \infty} \frac{\frac{d}{\mathrm{dx}} \left(\ln x\right)}{\frac{d}{\mathrm{dx}} \left(x\right)} = {\lim}_{x \to \infty} \frac{\frac{1}{x}}{1} = {\lim}_{x \to \infty} \frac{1}{x} = 0$

The limit approaches $0$ because $1$ divided over something approaching $\infty$ becomes closer and closer to $0$

For example, consider:

$\frac{1}{10} = 0.1$

$\frac{1}{100} = 0.01$

$\frac{1}{10000} = 0.0001$

We can see that as the denominator gets larger and larger, approaching $\infty$, the value gets smaller and smaller and more closer to $0$.

Nov 5, 2017

${\lim}_{x \to \infty} \ln \frac{x}{x} = 0$

#### Explanation:

The question is to find the value of $\ln \frac{x}{x}$ where $x \to \infty$

If we let $x = \infty$ then $\ln \frac{x}{x} = \frac{\infty}{\infty}$$=$ Undefined

So now we can apply L'Hospital's rule by differentiating the numerator and denominator individually.

So, $\frac{d}{\mathrm{dx}} \ln x = \frac{1}{x}$ and $\frac{d}{\mathrm{dx}} x = 1$

${\lim}_{x \to \infty} \frac{\frac{1}{x}}{1}$

${\lim}_{x \to \infty} \frac{1}{x}$

Now we let $x = \infty$

Therefore $\frac{1}{\infty} = 0$ because $\infty$ is a very large number and $1$ divided by a very large number always approaches $0$ and it is very close to $0$ but never quite gets there.