Question

Calculate the limit of the given function

Answer 1

Let's break down the fraction into multiple factors:

`\cfrac{\sin(ax)\cos(bx)}{\sin(cx)} = \sin(ax)\cdot\cos(bx)\cdot \cfrac{1}{\sin(cx)}`

Now we will manipulate the expression (multiply and divide by the same quantitues) in order to be able to use the known limit

`\displaystyle \lim_(x\to 0) \cfrac{\sin(x)}{x} = 1`

Here's the manipulated expression:

`\sin(ax)\cdot \cfrac{ax}{ax} \cdot \cfrac{1}{\sin(cx)}\cdot\cfrac{cx}{cx}\cdot\cos(bx)`

Rewrite the expression as

`\cfrac{\sin(ax)}{ax} \cdot \cfrac{cx}{\sin(cx)} \cdot \cfrac{ax}{cx} \cdot \cos(bx)`

The first two factors tend to 1, because that's the limit we mentioned before. The third factor is simply a/c, because the x's cancel out. Finally, we have

`\displaystyle \lim_(x\to 0) \cos(bx) = \cos(0)=1`

So, the final answer is

`\displaystyle \lim_(x\to 0) \cfrac{\sin(ax)}{ax} \cdot \cfrac{cx}{\sin(cx)} \cdot \cfrac{ax}{cx} \cdot \cos(bx) = 1 \cdot 1 \cdot \cfrac{a}{c} \cdot 1 = \cfrac{a}{c}`