# limit

1. ### Some limits manipulation.

Could someone explain the steps involved in this limit? I'm reading a proof and kinda stuck here. For k \in \mathbb{Z}^+ and c a positive constant, set \beta_k = c \sqrt{\frac{k}{\ln(k)}} and \delta_k = \frac{\ln(\frac{k}{\beta_k})}{k}. Then \lim\limits_{k \to \infty} (\beta_k(1-\delta_k)^k +...
2. ### Limit,sequence

Evaluate L=\lim_{n\rightarrow \infty} \dfrac{n^{n^2 +n +1}}{{e^{n^2 }(1^1 \cdot 2^2 \cdot \dotsc \cdot n^n ) }}.
3. ### limit again

Question: Evaluate \lim_{n\to+\infty}\frac{n^{2+p}}{2^n}, where p>1. Both denominator and numerator approach infinity. I used L'HÃ´pital's Rule, but unsuccessful. I used the equalities n^{2+p}=e^{(2+p)\ln{n}} and 2^n=e^{n\ln{2}}, but unsuccessful. Thank you.
4. ### A limit with factorials

Evaluate \lim_{n\rightarrow \infty }n!^{-n!}\cdot n^{n^{n}} .
5. ### Limit of inscribed regular polygons

An equilateral triangle of side 1 is inscribed by the largest square possible which is inscribed by the largest regular pentagon possible, ad infinitum. What is the radius of the limiting circle?
6. ### A limit

Evaluate \lim_{x\rightarrow 0 } \frac{x^{x+1} -\sin(x)}{x-\tan(x)}.
7. ### Basic limit with exponent

How can I prove the equality ? \lim_{k\rightarrow \infty} r^{k}=\lim_{k\rightarrow \infty } |r^{2}-r|^{k^2 }\: ; for 0< r \neq 1 .
8. ### Limit

How to show that \lim_{x\to\infty} ((3(x+1)^2-2)/(5(x+1)^2+7))^{(x+1)}/((3x^2-2)/(5x^2+7))^x=3/5? I used the fact that a^b=e^{b\ln{a}} but unsuccessful. Please give some hints. Thank you.
9. ### Cannot evaluate the limit

Evaluate without l'hopital's rule . L=\lim_{t\rightarrow 0} \frac{t-arctan(t)}{t^3 }.
10. ### Limit of postive function with bounded derivative and convergent improper integral

Is it true that for a continuous function X(t) where X(t)â‰¥0 and \int_{-\infty}^{-T}X(s)ds<\infty and X(t) is bounded together with its derivative: \lim \limits_{tâ†’âˆ’\infty} X(t)=0? If so, why?
11. ### A limit

Evaluate \lim_{n\rightarrow \infty }n^{-1} \underbrace{\ln \ln...\ln}_{n}(n).
12. ### Limit with zeta function

Evaluate \lim_{s\rightarrow \infty } \zeta(s)\cdot \zeta(2s)\cdot \zeta(3s)\cdot ...\cdot \zeta(s^2 ) .
13. ### Hard Limit

Evaluate the limit without L'hÃ´pitalâ€™s rule. \lim_{y\rightarrow \infty }\frac{\ln(yn)}{y^n } \; , n\in \mathbb{N}.
14. ### Limit with sequence

Simple Limit Evaluate \lim_{x\rightarrow 2 } \frac{2^x - x^2 }{2x-\sqrt{8x}}.
15. ### Evaluate the limit

\lim_{n\rightarrow \infty } \frac{1+1/2 +...+1/n }{\ln(n)}.
16. ### Central Limit Theorem for weighted summation of random variables?

Here is a quick question:- If X1, X2, X3,.... X20 are 20 random variables (independent/ idd) What would be the result of: 2*X1+5*X2+1*X3+18*X4...+0.5*X20? (linear combination of the random variables, with fixed known constants). Will the above function form a normal distribution if we...
17. ### Proof of e^x, lnx and e as limit without circular dependency?

People prove the derivative of e^x, \ln(x) using either the formula of e = \lim_{x \to 0} (1+x) ^ \frac{1}{x} or if they know one of the derivatives e^x, \ln(x) they use it to prove the other. My textbook first gave me without proof that \frac{d}{dx}e^x = e^x Then it found the limit of \ln(x)...
18. ### Limit of f(x)/q(x) with q(c) = 0 that can not be simplified.

Is it true that if \lim_{x \to c} \frac{f(x)}{q(x)} with q(c) = 0, f(c) \in R and if that fraction can not be simplified any more, then this limit will always diverge. Example: \lim_{x \to 1} \frac{1}{x-1} diverges because: \lim_{x \to 1^-} \frac{1}{x-1} = - \infty \lim_{x \to 1^+}...
19. ### Find the limit of f(t) using its graph.

Exercise: In question c, the answer is does not exist because "As t approaches 0 from the left, f(t) approaches -1 and as t approaches 0 from the right, f(t) approaches 1". The solution above is from Thomas-Calculus 13th edition. I can not understand the order... Why f(t) approaches -1 from...
20. ### Logarithmic Limit Problem

Lim x--> 0+ if (ln(xlna))(ln(ln(ax)/ln(x/a)) =6 find a