The value of k, for which the function f(x) = cases ( kx)/x + 3 x, & x ≠ 0 \\ 7, & x = 0 cases is continuous at x = 0, is Option 1: 3 Option 2: 5 Option 3: 1 Option 4: 4

Answer: 4. Solution: For continuity: _(x 0) f(x) = f(0). _(x 0) ( kx)/x = k and _(x 0) 3 x = 3. So k + 3 = 7, giving k = 4.

Calculus

Continuity & Differentiability

Easy

The value of k, for which the function $f(x) = \begin{cases} \frac{\sin kx}{x} + 3\cos x, & x \neq 0 \\ 7, & x = 0 \end{cases}$ is continuous at $x = 0$ , is

$3$

$5$

$1$

$4$

✅ Correct Option: 4

For continuity: $\lim_{x \to 0} f(x) = f(0)$ .

$\lim_{x \to 0} \frac{\sin kx}{x} = k$ and $\lim_{x \to 0} 3\cos x = 3$ .

So $k + 3 = 7$ , giving $k = 4$ .

2022: 16 July Shift 2

Match List I with List II

List I	List II
A. If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$ , then $\frac{g(x)}{f(x)}$ is	I. discontinuous at exactly three points.
B. The function $f(x) = \frac{4-x^2}{4x-x^3}$ is	II. continuous everywhere
C. The function $f(x) = \lvert x \rvert + \lvert x-1 \rvert$ is	III. discontinuous at $x = 0$ .
D. The function $f(x) = \lvert \sin x \rvert$ is	IV. continuous at $x = 0$ and $x = 1$

Choose the correct answer from the options given below:

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