I need some help in a problem from Calculus 1 class:
Find the lim sin3t + 4t
tsect
when t approaches 0.
To learn how to format using only text, try here. My guess as to your meaning, using LaTeX formatting, is as follows:
. . . . .$\displaystyle \lim_{t \rightarrow 0}\, \frac{\sin(3t)\, +\, 4t}{t \sec(t)}$
A good way to start might be to split the addition into two parts, also converting the reciprocal of the secant into a cosine:
. . . . .$\displaystyle \lim_{t \rightarrow 0}\, \frac{\sin(3t)\, cos(t)}{t}\, +\, 4\cos(t)$
You learned an important trig limit, probably when you were learning trig derivatives:
. . . . .$\displaystyle \lim_{\theta \rightarrow 0}\, \frac{\sin(\theta)}{\theta}\, =\, 1$
With this in mind, restate the limit as:
. . . . .$\displaystyle \lim_{t \rightarrow 0}\, 3\left(\frac{\sin(3t)}{3t}\right) \cos(t)\, +\, \lim_{t \rightarrow 0}\, 4\cos(t)$
The cosine can be simply evaluated, and the sine fraction is the limit they gave you earlier.