Question

For which pair of functions f(x) and g(x) below will the lim f(x)g(x)≠0 as x->infinity

Answer 1

f(x) = 10x + e^(-x); g(x) = 1/5x
f(x) = x^2; g(x) = e^(-4x)
f(x) =(Lnx)^3; g(x) = 1/x
f(x) = sqrt(x) ; g(x) = e^(-x)

Answer 2

is g(x)=(1/5)x or g(x)=1/(5x) ?

Answer 3

oh sorry it's g(x)=1/(5x)

Answer 4

Yes, it's fine...

Answer 5

\(\color{#000000 }{ \displaystyle \lim_{x\to\infty} f(x)\times g(x) }\)

Answer 6

What limit will this give you if you use the first pair?

Answer 7

limx→∞f(x)×1/(5x)?

Answer 8

yes, and you forgot to substitute the f(x).

Answer 9

\(\color{#000000 }{ \displaystyle \lim_{x\to\infty} \left[(10x + e^{-x})\times\frac{1}{5x}\right] }\)

Answer 10

limx→∞ 10x + e^(-x) ×1/(5x)

Answer 11

please expand in the limit

Answer 12

10x • 1/(5x) = ?
e\(^{-x}\) • 1/(5x) = ?

Answer 13

10x • 1/(5x) = 2
e−x • 1/(5x) = e^(-x)/(5 x)

Answer 14

((using the rules of exponents))

Answer 15

yes, and
e\(^{-x}\) / (5x) = 1 / (5x e\(^x\))

Answer 16

So our limit is

\(\color{#000000 }{ \displaystyle \lim_{x\to\infty} \left[2 + \frac{1}{5xe^x}\right] }\)

Answer 17

And since,


\(\color{#000000 }{ \displaystyle \lim_{x\to\infty} \left[2 + \frac{1}{5xe^x}\right]=\lim_{x\to\infty} \left[2 \right]+\lim_{x\to\infty} \left[ \frac{1}{5xe^x}\right] }\)

you should tell me what the limit will equal.

Answer 18

(evaluate each limit individually and add the limits)

Answer 19

did we say the limit is 0?

Answer 20

\(\color{#000000 }{ \displaystyle\lim_{x\to\infty} \left[(10x + e^{-x})\times\frac{1}{5x}\right] = \lim_{x\to\infty} \left[2 + \frac{1}{5xe^x}\right] \\ \displaystyle =\lim_{x\to\infty} \left[2 \right]+\lim_{x\to\infty} \left[ \frac{1}{5xe^x}\right] =2+0=2}\)

Answer 21

oh, I may be wrong. I am sorry. :)

Answer 22

\(\color{#000000 }{ \displaystyle \lim_{x\to a}(b)=b }\)

Answer 23

A function y=b will give you an output y=b for all x, wouldn't it?

Answer 24

Ans same with a limit, when you take
\(\color{#000000 }{ \displaystyle \lim_{x\to \infty }(2) }\)

Answer 25

chris, do you want to check other limits?

Answer 26

((A is the correct answer - and the only one, if this is not "for all that applies" question.))

Answer 27

Thank you!!!!!