Question

1.(sint)/t

2.(1 - cos t)/t

3.(e ^ t - 1)/t

Answer 1

Suppose that the task is to compute the limits of these functions as $t\rightarrow0$.

1. $\lim_{t\rightarrow 0}\frac{sin\, t}{t}.$ It is a well-known limit. It is possible to compute this limit by using geometric considerations or by using L'Hôpital's rule. Namely, $(sin\,t)'=cos\,t$ and $t'=1$, where $'$ denotes the derivative. Therefore, $\lim_{t\rightarrow 0}\frac{sin\, t}{t}=\lim_{t\rightarrow 0}\frac{cos\, t}{1}=1$
2. $\lim_{t\rightarrow 0}\frac{1-cos\, t}{t}$. Using trigonometric formulae, one receives: $1-cos\,t=2\,sin^2\,\frac{t}{2}$. We use the limit from the previous exercise and get: $\lim_{t\rightarrow 0}\frac{1-cos\, t}{t}=\lim_{t\rightarrow 0}2\frac{sin^2\, \frac{t}{2}}{\frac{t^2}{4}}\frac{t}4=0.$
3. $\lim_{t\rightarrow 0}\frac{e^t-1}{t}.$ It is also a well-known limit. It is possible to compute the limit using different considerations. We use L'Hôpital's rule: $(e^t-1)'=e^t,\quad t'=1$. Therefore: $\lim_{t\rightarrow 0}\frac{e^t-1}{t}.=\lim_{t\rightarrow 0}e^t=1.$