Exponential Indeterminate Forms
Read this topicExplanations 2
- Cases of the form f(x)g(x)f(x)^{g(x)}f(x)g(x) are strange because they are often not seen as indeterminate. 000^000 is generally assigned an integer value, 111, whereas forms su…
By Jack Gurev, Alex Federspiel
- In this exercise we prove that for any positive reals a1,…,ana_1,\ldots,a_na1,…,an, we have limx→0(a1x+⋯+anxn)1/x=(a1⋯an)1/n\lim_{x\to0} \left(\frac{a_1^x+\cdots+a_n^x}{n}\ri…
By Victor Wang, Moor Xu
Problems 6
- Compute limx→0(1+x)1/x\lim_{x\to0} (1+x)^{1/x}limx→0(1+x)1/x.
- Find the limit limx→1xx/sin(x−1)\lim_{x\to 1} x^{x/\sin(x-1)}limx→1xx/sin(x−1). Source: HMMT February 1998
- Compute limx→1+x21−x\displaystyle \lim_{x \to 1^+} x^{\frac2{1-x}}x→1+limx1−x2.
- Compute limx→0+(1x)x2\displaystyle \lim_{x\to0^+} \left( \frac1x \right)^{x^2}x→0+lim(x1)x2.
- 1∞=?1^{\infty} = ?1∞=?
- Suppose that each repetition succeeds with probability 99.999%99.999\%99.999%, independent of the success or failure of all other repetitions. What is the probability that the m…