LimitsMedium

Notable limit with sin(ax)/(bx)

Calculate lim_{x→0} sin(4x)/(2x) using the fundamental notable limit.

Given data

lim_{x→0} sin(4x)/(2x)

Steps

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Rewrite the expression to reveal the notable limit sin(t)/t → 1.
Write your working — the numerical result must appear in the text (coefficiente)
What is the limit of sin(4x)/(4x) as x→0?
Therefore the total limit equals?

Full worked solution

Rewrite the expression to reveal the notable limit sin(t)/t → 1.
$\dfrac{\sin(4x)}{2x} = \dfrac{4}{2} \cdot \dfrac{\sin(4x)}{4x}$
$\frac{\sin(4x)}{2x} = \frac{4}{2} \cdot \frac{\sin(4x)}{4x} = 2 \cdot \frac{\sin(4x)}{4x}$ . We factor the expression to reveal the fundamental notable limit $\sin(t)/t \to 1$ .
What is the limit of sin(4x)/(4x) as x→0?
$\lim_{t\to 0} \dfrac{\sin t}{t} = 1$
Let $t = 4x$ . As $x \to 0$ , $t \to 0$ . The fundamental notable limit gives $\displaystyle\lim_{t\to 0} \frac{\sin t}{t} = \mathbf{1}$ .
Therefore the total limit equals?
$2 \cdot 1 = 2$
$\displaystyle\lim_{x\to 0} \frac{\sin(4x)}{2x} = 2 \times 1 = \mathbf{2}$ . The coefficient 2 multiplied by the notable limit gives the result.

Result:lim_{x→0} sin(4x)/(2x) = 2.

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