Prove that f(x)=c has a solution

[sergey_le]

Homework Statement

Let function ƒ be f(x)=1/x+sin^2(x) and inff(0,∞)=0.
Prove that every c> 0 has a solution to the equation f(x)=c

Relevant Equations

-

I'm not sure that inff(0,∞)=0 can help But that was the first section of the question so I thought to point it out anyway.
I'm not sure what I'm supposed to do or what I'm supposed to show.
I was thinking of using the right environment of 0 where f aims for infinity but I don't know how it helps me.

 

[member 587159]

Hint: Show that if $x \to 0+$, then $f \to +\infty$. Since $f$ is continuous on $]0,\infty[$, the intermediate value theorem says that $f(x)=c$ must have a solution for $c > 0$.

You can do something similar to show that $\inf f((0, + \infty)) = 0$.

 

[sergey_le]

Hint: Show that if $x \to 0+$, then $f \to +\infty$. Since $f$ is continuous on $]0,\infty[$, the intermediate value theorem says that $f(x)=c$ must have a solution for $c > 0$.

You can do something similar to show that $\inf f((0, + \infty)) = 0$.

Thank you.