On Infinite Limits I am currently learning about infinite limits in Calculus, basically determining the limit of a function as x approaches infinity. However, I am struggling to understand the method being used to find it. 

Let’s take the function above. The method above seems to be to ignore all the terms with a lower power except the terms with the highest power. Then, because both of the highest terms are x^5, we cross them out and then divide their coefficients, to get a limit of 2/3. 
I guess you can do this because at infinity, the value of x^5 would be so big that it would dominate all the other values However, I still have a few problems with this method:


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*But on that logic, why should we care that the coefficient of the numerator of is 4 and the denominator’s coefficient is 6? The value of x^5 is so big that it would dominate both of them anyway? At this rate, because it would dominate everything, both the numerator and denominator of every function approaching infinity should be infinity over infinity! So wouldn’t the limit of all infinite functions be 1? 





*This is another example that confuses me. Apparently, when x is infinity, you can ignore the 10, because infinity would dominate the whole function, and therefore the limit would be 0. But this doesn’t make sense to me! Even at infinity, the difference between the 2 would be 10, not 0. No matter how large a number you sub in, the difference between the 2 functions will be 10, and therefore, how can the functions approach 0 as x approaches infinity? 

*I also I get that if you zoom out in the function above, it would truly seem like that function is approaching 0.  But then that wouldn’t be the limit of the function would it? That would be zooming out! Once we zoom back in, we will be able see that the function is sticking at 10, not getting closer to 0! So how can we say that the limit of the function at infinity is 10? 
Can someone explain the above to me? Can you also not make the explanation too rigorous? I’m just learning Khan Academy Calculus, and still haven’t touched on things like epsilon delta proofs yet. Thank you! 

For 2 and 3, it seems like you are making an algebra mistake.
$$(\sqrt x+10)^2=x+20\sqrt x+100\ne x+100$$
For example, if $x=10000, \sqrt x=100$.  $\sqrt{10100}<101$.  Indeed, your problem shows $\sqrt{100+x}-\sqrt x=\dfrac{100}{\sqrt{100+x}+\sqrt x}$.  As x increases, the denominator increases without bound while the numerator stays the same.
