# Thread: Surjuctivity of a function

1. ## Surjuctivity of a function

Why is the following function surjective(onto) ?

$f: (1,\infty)\longrightarrow (\ln 2,\infty)$
$f(x)=\int_{x^2}^{x^4}\! \frac{1}{\ln t}\, dt$

2. Originally Posted by bigli
Why is the following function surjective(onto) ?

$f: (1,\infty)\longrightarrow (\ln 2,\infty)$
$f(x)=\int_{x^2}^{x^4}\! \frac{1}{\ln t}\, dt$
Try using the IVT and creating some nice inequalities

3. is this IVT? : intermediate value theorem. anyway, How do I use it?

4. Originally Posted by bigli
What is IVT?
Intermediate value theorem.

Here, f(x) is $\int_{x^2}^{x^4}\frac{1}{ln t} dt$ which is continous on [tex][1, \infty) because ln t is continuous for all positive t and the integral of a continuous function is continuous.
7. In fact, how do I show $\lim_{x\rightarrow 1^+ }\int_{x^2}^{x^4}\!\frac{1}{\ln t}\, dt=\ln 2$ ?