Pair of functions that satisfies the limit

Question

For which pair of functions f(x) and g(x) below will the  ?

Question options:

1)	f(x) = ex; g(x) = x3
2)	f(x) = x5; g(x) = ex
3)	f(x) = x3; g(x) = ln(x)
4)	f(x) = x-2; g(x) = e-x

Answer 1

Hi Marietta T,

So you are learning how to gauge the behavior of functions as the argument goes to infinity.

In particular, you are comparing that behavior for polynomial functions (such as x^3) and for exponential functions (such as e^x).

The essential concept here (as it is for limits as x->0), is to start comparing the slopes of the functions, i.e. their first derivatives (i.e. form the ratio of their first derivatives). If that doesn't give you a zero value or an infinite value, then you take their second derivatives as a ratio, and so on.

So, as an example consider the most basic polynomial function: f(x)=x. As x->.infinity., the ratio of f(x+1)/f(x) = f'(x)/x goes towards 1. Why is that? B/c that additional proportional "+1" becomes insignificant with respect to that "x" in the denominator. So what does that mean with respect to limits at infinity and exponential functions? The exponential function equivalent ratio is f(e^(x+1))/f(e^x) = e at all times, for all derivatives. So that ratio is a constant value -- e, and the particular function f', f'', f''' and so on is always just e^x. So that function -- f(x) = e^x -- is "bigger" than a power-of-x function -- "bigger" in the sense of lim as x->.infinity. I'll use "bigger" and "smaller" below in that sense (to distinguish between bigger and smaller, without quotes, which just represent the function values at a particular non-infinite value of x).

So, if there is a polynomial function -- even of high degree, like x^10000000, in the numerator, but a humble exponential function -- e^x -- in the denominator, the e^x will eventually be larger, so the limit of their ratio will go to zero.

ln(x), on the other hand, climbs slower and slower as x increases, so it's "smaller" than even just f(x)=x .

You may have an intuition regarding x^y, where 0<y<1. See if you can use your math to work out the case for x^0.5 vs. ln(x). Hint: if there is a zero ratio for lim( x:y), then there is a zero ratio for lim( x^2:y^2), and so on. You could work it out that way, or maybe you already just know that f'(ln(x)) = 1/x ,and you can take it from there.

For negative exponents, just flip them with respect to numerator&denominator and change the sign on the exponent, you know the drill. That will make a problem with just positive exponents, and you know how to do that.

When you encounter more complicated functions such as e^(x^2), just take things one step at a time, figuring out which function blows up faster. In general, if you end up with multiple terms in your derivatives for both numerator and denominator, only keep the "biggest" one for each (of numerator and denominator) for taking further derivatives. The "smaller" terms don't matter at all by comparison, so don't even bother to keep track of them.

-- Cheers, ----- Mr. d.