The Limit — Question 7

Question 7 - Solution

We are given a piecewise function: $$f(x) = \begin{cases} \dfrac{\sin(5x)}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$$

To make $f(x)$ continuous at $x = 0$, we must have: $$\lim_{x \to 0} f(x) = f(0) = k$$

We now compute: $$\lim_{x \to 0} \frac{\sin(5x)}{x}$$

We use the identity: $$\lim_{x \to 0} \frac{\sin(ax)}{x} = a$$

So: $$\lim_{x \to 0} \frac{\sin(5x)}{x} = 5$$

Thus, for continuity at $x = 0$, we must have: $$k = \boxed{5}$$

Conclusion: The function is continuous at $x = 0$ if $\boxed{k = 5}$.