This list consists of the limits that I found most challenging.

1. $$\lim_{n \to \infty} \left( \frac{n!}{n^n} \right) ^\frac 1n$$
2. $$\lim_{x \to 0} \left( \frac{1}{\ln(x + \sqrt{x^2+1})} - \frac 1{\ln(x+1)} \right)$$
3. $$\lim_{n \to \infty} \frac{n + n^2 …

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Simple limit problems consist of the form $\lim_{x \to 0}\frac{O_1(x)O_2(x)..}{O_a(x)O_b(x)..}$, such as $\lim_{x \to 0} \frac{\sin 3x \tan 2x \tan^{-1} 5x}{x^2 \ln(1+x)}$. Here, the infinitesimals are well defined and cancel out easily. Some …

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The USA currently stands at 1.76 million COVID-19 cases. That's more than the next 5 nations combined. A large number of these cases are due to government inaction against the virus. The lack of a concerted lockdown across the country is also to blame. Here's a graph showing the …

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The Previous COVID regression analysis was fairly accurate. However, the opening of lockdown offset the statistics a bit and now there are more number of projected cases. Here is a recomputation of the statistics, which projects an average of 172,000 cases by June 1 and 520,000 cases overall …

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This is an interesting number theory fact that seems strange when taken at face value. Here's a small proof of it:

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This is a regression analysis attempt for the COVID-19 spread data. The graphs represent total cases per day. The Orange graph is USA, the smaller graph on the left is China and the graph on the right is India. A Standard gaussian curve of the form $Ae^{-b(x-c)^2 …

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Define $ f(x) = ax^2 + bx + c , a,b,c \in \mathbb{R}$ and $ f^n(x) = f(f^{n-1}(x)), n>1 $. Prove that the real roots of $ f^n(x) $ are symmetric about the vertical line passing through vertex i.e. $ x = \frac{-b}{2a} $

This seems like …

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Evaluate the integral $$\int_0^1 \frac{\log(x)}{x-1}dx$$

There are a few methods of doing this: the first one uses the taylor series expansion of \( \log(1-x) \): $$I = -\int_0^1 \frac{\log(1-x)}{x}dx$$ $$I = \int_0^1 \frac 1x \left( x + \frac{x^2}{2} + \frac{x …

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A speculative format for the 2018 ICSE Mathematics paper, created by analysis of the mathematics question papers of the past 23 years.

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