Problem 594
Which one is greater? \(\displaystyle 5^{\log_73}\) or \(\displaystyle 3^{\log_75}\)? Difficulty: 1. |
Problem 596
Prove that \(\displaystyle \displaystyle\lim_{x\to\infty} \frac{\ln x}{x} = 0\) and \(\displaystyle \displaystyle\lim_{x\to+0} x\cdot\ln x = 0\). Difficulty: 3. |
Problem 597
\(\displaystyle \lim_{x\to+0} x^x = ? \qquad \lim_{x\to+\infty} \sqrt[x]{x} = ? \) Difficulty: 4. |
Problem 611
Prove that if \(\displaystyle x>0\), \(\displaystyle n\in\N\) then \(\displaystyle e^x > 1+\sum_{k=1}^n\frac{x^k}{k!}. \) Difficulty: 4. |
Problem 612
\(\displaystyle \lim_{x\to \infty}\frac{x^2-\sqrt{x^3+1}}{\root{3}\of{x^6+2}-x}=? \) Difficulty: 4. |
Problem 613
\(\displaystyle \lim_{x\to\infty} \frac{\sqrt{2^x+3^x}+4^x}{\displaystyle\bigg(1+\frac1x\bigg)^{x^2}} = ? \) Difficulty: 4. |
Problem 614
\(\displaystyle \lim_{x\to+0} e^{\ln x/(\ln|\ln x|)} = ? \) Difficulty: 4. |
Problem 595
Suppose that \(\displaystyle \phi>0\), and \(\displaystyle \log \phi\) is convex. Prove that \(\displaystyle \phi\) is convex and show that the reverse implication does not hold. Difficulty: 5. |
Problem 610
\(\displaystyle \lim_{x\to-0} \left(1+\frac1x\right)^{x} = ? \) Difficulty: 5. |
Problem 616
Prove that \(\displaystyle \displaystyle\ln(n+1)<1+\frac12+\frac13+\ldots+\frac1n \le (\ln n)+1\). Difficulty: 5. |
Problem 603
\(\displaystyle \lim_{x\to+0} \left(1+\frac1x\right)^{x} = ? \) Difficulty: 6. |
Problem 604
Prove that if \(\displaystyle 0<x\), \(\displaystyle x\ne1\) then \(\displaystyle \ln x< x-1\). Difficulty: 6. |
Problem 605
Prove that if \(\displaystyle 0<x<1\) then \(\displaystyle \ln(x)>1-\dfrac1x\). Difficulty: 6. |
Problem 593
Prove that if \(\displaystyle f:\R\to(0,\infty)\) is continuous and for all \(\displaystyle x,y\in\R\) the equality \(\displaystyle f(x+y)=f(x)\cdot f(y)\) holds then \(\displaystyle f\) is an exponential function. Difficulty: 7. |
Problem 599
Prove that for the reals \(\displaystyle 0<a<b\) the equality \(\displaystyle a^b=b^a\) holds if and only if there is a positive number \(\displaystyle x\) for which \(\displaystyle a=\left(1+\frac1x\right)^x\) and \(\displaystyle b=\left(1+\frac1x\right)^{x+1}\). Difficulty: 7. |
Problem 607
Find reals \(\displaystyle a,b\) such that for all \(\displaystyle |x|<\frac12\) we have \(\displaystyle 1+x+ax^2<e^x<1+x+bx^2\). Difficulty: 7. |
Problem 608
Find reals \(\displaystyle a,b\) such that for all \(\displaystyle |x|<\frac12\) we have \(\displaystyle x+ax^2<\ln(1+x)<x+bx^2\). Difficulty: 7. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |