Limit of infinitely small oscillating functions I have a question about limit of "small" oscillating functions.
Let's say I have the two functions of the image (really don't know what they could be). Is the limit for x that goes to infinity of the one in "black" 1 as the one in "red"?



Because if we have a function like f(x)=sin(x)*(1/x) it is clear that the limit for x that goes to infinity is 0, as it goes up and down of always smaller amounts around 0.
But of the function i drawn things are a bit different, as it goes only above y=1 to then "touch" y=1 back again. So I'm not sure if the limit is definite.
$$\lim_{x\to\infty}\frac{\sin^2(x)+\sqrt{x^2+1}}{x}=1.$$
Yes it’s possible.... let assume a function f(x) which is the upper bound of our oscillating function h(x).we see that 1&lt;=h(x) &lt;=f(x) 
So here if f(x) convergence to 1 then h(x) must have to converge at 1. But if the oscillating function h(x) convergence to 1 we can’t say anything about f(x). 
Thank you