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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.

Calculus Warmups: Level 3 Challenges

         

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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