# How do you evaluate [sin(3t) + 4t] / [t sec t] as t approaches 0?

Oct 24, 2017

${\lim}_{t \rightarrow 0} \frac{\sin \left(3 t\right) + 4 t}{t \sec \left(t\right)} = 7$

#### Explanation:

This is tricky.

Simplify $\frac{\sin \left(3 t\right) + 4 t}{t \sec \left(t\right)}$ into $\frac{\cos \left(t\right) \cdot \left(\sin \left(3 t\right) + 4 t\right)}{t}$

Then apply some addition formulas and double-angle formulas.

Firstly let $\sin \left(3 t\right) = \sin \left(2 t + t\right)$
$= \sin \left(2 t\right) \cdot \cos \left(t\right) + \cos \left(2 t\right) \cdot \sin \left(t\right)$.

Then expand this with two compound angle formulas:

$\sin \left(2 t\right) \cdot \cos \left(t\right) + \cos \left(2 t\right) \cdot \sin \left(t\right)$
$= 2 \sin \left(t\right) \cdot \cos \left(t\right) \cdot \cos \left(t\right) + \left(2 {\cos}^{2} \left(t\right) - 1\right) \cdot \sin \left(t\right)$.
$= 2 \sin \left(t\right) \cdot {\cos}^{2} \left(t\right) + 2 {\cos}^{2} \left(t\right) \cdot \sin \left(t\right) - \sin \left(t\right)$
$= 4 \sin \left(t\right) \cdot {\cos}^{2} \left(t\right) - \sin \left(t\right)$.

Then add the $4 t$:
$4 \sin \left(t\right) \cdot {\cos}^{2} \left(t\right) - \sin \left(t\right) + 4 t$.

Then multiply by $\cos \left(t\right)$ so finally the original numerator is:
$4 \sin \left(t\right) \cdot {\cos}^{3} \left(t\right) - \sin \left(t\right) \cdot \cos \left(t\right) + 4 t \cdot \cos \left(t\right)$.

So, to find:
${\lim}_{t \rightarrow 0} \frac{4 \sin \left(t\right) \cdot {\cos}^{3} \left(t\right) - \sin \left(t\right) \cdot \cos \left(t\right) + 4 t \cdot \cos \left(t\right)}{t}$

I shall steer clear of fancy formulas, and rely on an intuitive approach, if this suits.

Set the following values:
The approach of $t$ to zero can be expressed as $t = \frac{1}{\infty}$.
Likewise, as $t$ approaches zero, $\sin \left(t\right) = \frac{1}{\infty}$ and $\cos \left(t\right) = 1$.

Hence, remove any $\cos$ terms from the expression.

So the numerator becomes:
$4 \sin \left(t\right) \cdot {\cos}^{3} \left(t\right) - \sin \left(t\right) \cdot \cos \left(t\right) + 4 t \cdot \cos \left(t\right)$ $= \frac{4}{\infty} - \frac{1}{\infty} + \frac{4}{\infty}$
$= \frac{7}{\infty}$

Finally, dividing by the $t$ in the denominator:

$\frac{\frac{7}{\infty}}{\frac{1}{\infty}} = 7$.