Question

Determine the limit of the function (if it exists). lim (sin 2x)/(sin 3x) as x--> 0.

Answer 1

\(\bf lim_{x\to 0} \cfrac{sin(2x)}{sin(3x)}\\ \quad \\ \quad \\
\cfrac{sin(2x)}{sin(3x)} \times \cfrac{3x}{3x} \implies \cfrac{3x}{sin(3x)}\cdot \cfrac{sin(2x)}{3x}\\ \quad \\ \quad \\
\textit{now, if we make say } 2x = u \implies x = \cfrac{u}{2}\\ \quad \\ \quad \\
\cfrac{3x}{sin(3x)}\cdot \cfrac{sin(2x)}{3x} \implies 1 \cdot \cfrac{sin(2x)}{3x} \implies \cfrac{sin(2x)}{3x}\\ \quad \\ \quad \\
\implies \cfrac{sin(u)}{3\cdot\frac{u}{2}}\implies \cfrac{sin(u)}{\frac{3}{2}\cdot u} \implies \cfrac{sin(u)}{u} \cdot \cfrac{1}{\frac{3}{2}}
\)

sorry it took me a bit
see the liimit?

Answer 2

this makes sense but on my paper it says hint: find lim\[(2\sin 2x/2x)(3x/3\sin 3x)\] as x-->0. so idk what to do w/ that

Answer 3

right.... if I recall correctly the procedure is the same though, but yes, that's the idea

Answer 4

okay thanks :)

Answer 5

is pretty much the same

Answer 6

yw