calculus question

Question

Which of the following functions grows the fastest as x goes to infinity? (4 points)

	2x
	ln(x)
	sin(x)
	x20

Answer 1

The first step to this problem is to realize that since they are asking you about how fast the functions are growing, you'll be dealing with limits of the derivatives rather than limits of the functions themselves. Thus all the limits we're dealing with are:

1. ^lim_x→∞ ^{(ln2)(2^x)}
2. ^lim_x→∞ ^(1/x)
3. ^lim_x→∞ ^cos(x)
4. ^lim_x→∞ ^{20x^19}

^{I wrote these in the same order that the original functions are written in, so the last one corresponds to the last function written in the question etc. With these limits, we are looking to see which one has the largest value since the values of the limits correspond to the rates of change of the original functions at infinity.}

^{In the first limit, the limit, and therefore the rate of change for the original function, will go to infinity since any number greater than one to the infinite power will go to infinity.}

^{In the second limit, the limit will go to zero since any finite number divided by infinity is zero. Therefore the rate of change as x goes to infinite for this function is zero.}

^{In the third limit, technically we would say the limit does not exist because cosine is an oscillating function. However, cosine always has a value between -1 and 1, so we can put the limit somewhere between there and since this is less than infinity we know that the rate of change is greater in limit 1, so we can go ahead and eliminate limit 3 as a contender.}

^{In the third limit, the limit will go to infinity since infinity to any finite positive power is infinity.}

Initially these results appear confusing. Both limit 1 and limit 4 reach infinity. However, limit 1 is an exponential function while limit 4 is a polynomial. Exponential functions always have a higher rate of increase at infinity than polynomials. The best way to explain this is by explaining that for an exponential function a^x, f(n+1) is a number of times greater than f(n) equal to a, which is 2 in the case of limit 1. However with a polynomial f(n+1) will be of a relatively comparable value to f(n) at extremely large values for n. This is evident even for pretty small numbers. For example lets take 10000 to the 19th power and we get 1 x 10^76. Then lets take 10001 to the 19th and we get 1.001 x 10^76. Whereas with an exponential function 2 to the 10001 will be twice as large as 2 to the 10000. Therefore we can say that at extremely large numbers such as infinity, exponential functions increase faster. So the function that grows the fastest as x approaches infinity is 2^x.