Question

estimate the limit numerically or state that the limit does not exist (dne): \\(\lim\limits_{x \to 0} \frac{\sin(14x)}{x}\\) give your answer to at least three decimal places question help: video 1 video 2 submit question

Explanation:

Step1: Recall standard limit identity

We use the known limit $\lim_{h \to 0} \frac{\sin(h)}{h} = 1$

Step2: Rewrite the given expression

Let $h = 14x$. As $x \to 0$, $h \to 0$. Rewrite the limit:
$$\lim_{x \to 0} \frac{\sin(14x)}{x} = \lim_{x \to 0} \frac{\sin(14x)}{x} \cdot \frac{14}{14} = 14 \cdot \lim_{x \to 0} \frac{\sin(14x)}{14x}$$

Step3: Apply standard limit

Substitute $h=14x$ into the standard limit:
$$14 \cdot \lim_{h \to 0} \frac{\sin(h)}{h} = 14 \cdot 1$$

Answer:

14.000