How to find lim(ln(2x)/(2+x)) x->  infinity?

Sep 3, 2017

The limit equals $0$

Explanation:

Since we are of the form $\frac{\infty}{\infty}$, we can use L'Hospitals rule.

$L = {\lim}_{x \to \infty} \frac{\frac{2}{2 x}}{1}$

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

This is now a recognizable and commonly seen limit.

$L = 0$

Hopefully this helps!

Sep 4, 2017

${\lim}_{x \rightarrow \infty} \ln \frac{2 x}{2 + x} = 0$

Explanation:

Logarithmic functions grow slower than polynomial functions. Polynomial functions grow slower than exponential functions.

Since $\ln \left(2 x\right)$ is logarithmic and $2 + x$ is a polynomial (it only has a degree of $1$, but it's still a polynomial), the polynomial in the denominator will grow faster than the logarithmic function in the numerator.

Thus, the denominator will outpace the numerator and as $x$ goes to infinity, the limit approaches $0$.

Note what would happen if the fraction were inverted:

${\lim}_{x \rightarrow \infty} \ln \frac{2 x}{2 + x} = 0$

${\lim}_{x \rightarrow \infty} \frac{2 + x}{\ln} \left(2 x\right) = \infty$