Use series to evaluate limit

2,734

We have

$$3x^2(e^{-2/x^2}-1)=3x^2\left(1-\frac{2}{x^2}+O\left(\frac1{x^4}\right)-1\right)=-6+O\left(\frac1{x^2}\right)\xrightarrow{x\to\infty}-6$$

Share:
2,734

Related videos on Youtube

Qwynn
Author by

Qwynn

Updated on August 01, 2022

Comments

  • Qwynn
    Qwynn over 1 year

    limit as x approach infinity of $3(x^2)(e^{-2/x^2}-1)$. I don't know what series to use.

    • André Nicolas
      André Nicolas over 9 years
      Let $x^2=\frac{1}{t}$. So we are interested in $\frac{3}{t}(e^{-2t}-1)$. Use the familiar Maclaurin expansion.
    • Qwynn
      Qwynn over 9 years
      Ah! Written that way you can see what to do right away! thank you
    • André Nicolas
      André Nicolas over 9 years
      You are welcome. Limits at $0$ make Maclaurin series natural, since these give the behaviour of a function near $0$.
  • Admin
    Admin over 9 years
    You're welcome.