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Calculus Warmups

The concepts of limits, infinitesimal partitions, and continuously changing quantities paved the way to Calculus, the universal tool for modeling continuous systems from Physics to Economics. See more

Level 3

         

Is \(f(x)= x + \sin x\) an injective function?

\[\large\sum _{ j=0 }^{ \infty }{ \sum _{ i=0 }^{ \infty }{ \frac { { 2 }^{ -(i+j) } }{ i+j+1 } } } = \ ?\]

Give your answer to 3 decimal places.

\[ \large \lim _{xy\to 1}\left(\frac{\ln\:x}{\ln\:y}+\frac{\ln\:y}{\ln\:x}\right)= \ ? \]

\[ \large 2 \times \lim_{h\to0} \dfrac{\int_1^{1+h} x^x \, dx - \int_{1-h}^1 x^x \, dx }{h^2} =\, ? \]

Some people claim that \( 0 ^ 0 = 1 \). What is

\[ \Large \lim_{ x \rightarrow 0^+ } x ^ { \frac{ 1}{\ln x} }?\]

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