The velocity in ft/sec of a falling object is modeled by \(r(t)=-\sqrt{32\over k}{1-e^{2t\sqrt{32k}}\over1+e^{-2r\sqrt{32k}}}\), where k is a constant that depends upon the size and shape of the object and the density of the air. Find the  limiting velocity of the object, that is, find \(lim_{t\rightarrow \infty}r(t).\)

Show that \(lim_{x\rightarrow\infty}{1+2+3+...+n\over 3n^2+7n+2}={1\over6}\)

Show that \(lim_{x\rightarrow\infty} {1\over 1.2}+{1\over 2.3}+{1\over 3.4}+...+{1\over n(n+1)}=1\)

Show that \(lim_{x\rightarrow 0^+}x[\left\lfloor {1\over x} \right\rfloor+\left\lfloor {2\over x} \right\rfloor +....+\left\lfloor {15\over x}\right\rfloor ]=120\)

Do the limits of following functions exist as x\(\rightarrow 0?\) State reasons for your answer.\(sin(x -\left\lfloor x \right\rfloor) \over x- \left\lfloor x \right\rfloor\)

Evaluate the following limits :\(lim_{\alpha\rightarrow 0}{sin (\alpha)^n\over( sin \alpha )^m}\)

Describe the interval(s) on which each function is continuous.
\(h(x)= \begin{cases}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{cases}\)

A tomato wholesaler finds that the price of a newly harvested tomatoes is Rs. 0.16 per kg if he purchases fewer than 100 kgs each day. However, if he purchases at least 100 kgs daily, the price drops to Rs. 0.14 per kg. Find the total cost function and discuss the cost when the purchase is 100 kgs.

Let \(f(x)= \begin{cases}0, & \text { if } x<0 \\ x^{2}, & \text { if } 0 \leq x<2 \\ 4, & \text { if } x \geq 2\end{cases}\).Graph the function. Show that f(x) continuous on\((-\infty,\infty)\).

Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f.