ContinuityMedium

Continuity with parameter — sin(kx)/x

Find k such that f(x) = { sin(kx)/x for x≠0, 2 for x=0 } is continuous at x=0.

Given data

f(x) = sin(kx)/x for x≠0f(0) = 2

Steps

0 / 3

Set up the continuity condition at x=0: the limit must equal f(0).
Write your working — the numerical result must appear in the text (condizione)
Calculate the limit as a function of k using the notable limit.
What must k equal?

Full worked solution

Set up the continuity condition at x=0: the limit must equal f(0).
$\lim_{x\to 0} \dfrac{\sin(kx)}{x} = 2$
For continuity at $x=0$ : $\displaystyle\lim_{x\to 0} f(x) = f(0) = 2$ . We need $\displaystyle\lim_{x\to 0} \frac{\sin(kx)}{x} = 2$ .
Calculate the limit as a function of k using the notable limit.
$\lim_{x\to 0} \dfrac{\sin(kx)}{x} = k \cdot \lim_{x\to 0} \dfrac{\sin(kx)}{kx} = k$
$\displaystyle\lim_{x\to 0} \frac{\sin(kx)}{x} = \lim_{x\to 0} k\cdot\frac{\sin(kx)}{kx} = k \cdot 1 = k$ . Using the notable limit $\sin(t)/t \to 1$ with $t = kx$ .
What must k equal?
$k = 2$
Setting $k = f(0) = 2$ gives $\mathbf{k = 2}$ . For $k = 2$ , the limit matches the function value, making $f$ continuous at $x = 0$ .

Result:k = 2. With this value the discontinuity is eliminated.

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