#2**+10 **

Here's another way to do this

tan 12t / sin4t =

(sin12t/cos12t) / sin4t = (divide numerator and denominator by t)

(sin12t/t) /[(sin4t/t)(cos12t)] = (multiply the numerator by 12/12 and the denominator by 4/4)

(12sin12t/12t) / [ (4sin4t/4t)(cos12t)]

Now

lim t → 0 (12sin12t/12t) = 12 and

lim t → 0 (4sin4t/4t) = 4 and

lim t → 0 (cos12t) = 1 so

12/(4 * 1 ) = 12/4 = 3

CPhill
Dec 7, 2014

#1**+10 **

$$\\\displaystyle\lim_{t\rightarrow 0}\: \;\frac{tan12t}{sin4t}\\\\\\

=\displaystyle\lim_{t\rightarrow 0}\: \;\frac{sin12t}{cos12t\;sin4t}\\\\\\

=\displaystyle\lim_{t\rightarrow 0}\: \;\frac{sin4tcos8t+cos4tsin8t}{cos12t\;sin4t}\\\\\\

=\displaystyle\lim_{t\rightarrow 0}\: \;\frac{sin4tcos8t+cos4t*2sin4tcos4t}{cos12t\;sin4t}\\\\\\

=\displaystyle\lim_{t\rightarrow 0}\: \;\frac{cos8t+cos4t*2cos4t}{cos12t}\\\\\\

=\frac{cos0+cos0*2cos0}{cos0}\\\\\\

=\frac{1+1*2*1}{1}\\\\

=3$$

Melody
Dec 7, 2014

#2**+10 **

Best Answer

Here's another way to do this

tan 12t / sin4t =

(sin12t/cos12t) / sin4t = (divide numerator and denominator by t)

(sin12t/t) /[(sin4t/t)(cos12t)] = (multiply the numerator by 12/12 and the denominator by 4/4)

(12sin12t/12t) / [ (4sin4t/4t)(cos12t)]

Now

lim t → 0 (12sin12t/12t) = 12 and

lim t → 0 (4sin4t/4t) = 4 and

lim t → 0 (cos12t) = 1 so

12/(4 * 1 ) = 12/4 = 3

CPhill
Dec 7, 2014