Answer to Question #320184 in Calculus for Faye

Question

Evaluate limits involving the expressions (sin t/t), (1 - cos t)/t) and ((e ^ t- 1)/t) and indeterminate forms type "0/0"

Directions: Create example of limits of functions and evaluate their limits.

1.(sint)/t

2.(1 - cos t)/t

3.(e ^ t - 1)/t

Answer 1

Using L'Hospital's rule we can write:

$\displaystyle\lim_{\mathclap{t\to0}} {\frac{\sin t}{t}}= \lim_{\mathclap{t\to0}}{\frac{(\sin t)’}{t’}}= \lim_{\mathclap{t\to0}} {\frac{\cos t}{1}}=1$

$\displaystyle\lim_{\mathclap{t\to0}} {\frac{1-\cos t}{t}}= \lim_{\mathclap{t\to0}}{\frac{(1-\cos t)’}{t’}}= \lim_{\mathclap{t\to0}} {\frac{\sin t}{1}}=0$

$\displaystyle\lim_{\mathclap{t\to0}} {\frac{e^t-1}{t}}= \lim_{\mathclap{t\to0}}{\frac{(e^t-1)’}{t’}}= \lim_{\mathclap{t\to0}} {\frac{e^t}{1}}=1$