Answers to Relative Rates of Growth Homework
For each pair of functions given, determine which one (if either) grows faster as x → ∞.
You must give detailed arguments using limits for #2c,d,e and #3a,b,c,e,g. For each of
the others you may use limits or just explain your reasoning with a sentence. Try to notice
patterns while you do this!
√
1.a) x2 , 2 − x + 4x2
same growth rate
3.a) ln x, log2 x
same growth rate
b) x2 , sin(x3 ) + x2
same growth rate
√
√
2
3
c) 100x , 2 − x + 4x 2 − x + 4x3 grows faster
b) ln x, x
√
c) ln x, 10 x
x grows faster
√
10
x grows faster
d) 100x2 , 2x + x2
d) ln x, cos x
ln x grows faster
2x + x2 grows faster
2
e) ln x, ln x
2.a) x100 , ex
b) 2x , ex
ex grows faster
c) 2x , e−x
2x grows faster
d) 22x , ex
22x grows faster
e) x2x , ex
f) x, ecos x
ex grows faster
ex grows faster
x grows faster
same growth rate
f) ln x, (ln x)2 (ln x)2 grows faster
g) ln x, ln(ln x) ln x grows faster