chris215
  • chris215
For which pair of functions f(x) and g(x) below will the lim f(x)g(x)≠0 as x->infinity
Mathematics
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SOLVED
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jamiebookeater
  • jamiebookeater
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chris215
  • chris215
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)
SolomonZelman
  • SolomonZelman
is g(x)=(1/5)x or g(x)=1/(5x) ?
chris215
  • chris215
oh sorry it's g(x)=1/(5x)

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More answers

SolomonZelman
  • SolomonZelman
Yes, it's fine...
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle \lim_{x\to\infty} f(x)\times g(x) }\)
SolomonZelman
  • SolomonZelman
What limit will this give you if you use the first pair?
chris215
  • chris215
limx→∞f(x)×1/(5x)?
SolomonZelman
  • SolomonZelman
yes, and you forgot to substitute the f(x).
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle \lim_{x\to\infty} \left[(10x + e^{-x})\times\frac{1}{5x}\right] }\)
chris215
  • chris215
limx→∞ 10x + e^(-x) ×1/(5x)
SolomonZelman
  • SolomonZelman
please expand in the limit
SolomonZelman
  • SolomonZelman
10x • 1/(5x) = ? e\(^{-x}\) • 1/(5x) = ?
chris215
  • chris215
10x • 1/(5x) = 2 e−x • 1/(5x) = e^(-x)/(5 x)
SolomonZelman
  • SolomonZelman
((using the rules of exponents))
SolomonZelman
  • SolomonZelman
yes, and e\(^{-x}\) / (5x) = 1 / (5x e\(^x\))
SolomonZelman
  • SolomonZelman
So our limit is \(\color{#000000 }{ \displaystyle \lim_{x\to\infty} \left[2 + \frac{1}{5xe^x}\right] }\)
SolomonZelman
  • SolomonZelman
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.
SolomonZelman
  • SolomonZelman
(evaluate each limit individually and add the limits)
SolomonZelman
  • SolomonZelman
did we say the limit is 0?
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle\lim_{x\to\infty} \left[(10x + e^{-x})\times\frac{1}{5x}\right] = \lim_{x\to\infty} \left[2 + \frac{1}{5xe^x}\right] \\ \displaystyle =\lim_{x\to\infty} \left[2 \right]+\lim_{x\to\infty} \left[ \frac{1}{5xe^x}\right] =2+0=2}\)
anonymous
  • anonymous
oh, I may be wrong. I am sorry. :)
SolomonZelman
  • SolomonZelman
\(\color{#000000 }{ \displaystyle \lim_{x\to a}(b)=b }\)
SolomonZelman
  • SolomonZelman
A function y=b will give you an output y=b for all x, wouldn't it?
SolomonZelman
  • SolomonZelman
Ans same with a limit, when you take \(\color{#000000 }{ \displaystyle \lim_{x\to \infty }(2) }\)|dw:1450381494548:dw|
SolomonZelman
  • SolomonZelman
chris, do you want to check other limits?
SolomonZelman
  • SolomonZelman
((A is the correct answer - and the only one, if this is not "for all that applies" question.))
chris215
  • chris215
Thank you!!!!!

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