Let . Then the number of solutions of the equation f(x) = sin x, is: [2026]

Question

Let $f (x) = \lim_{y \to 0} \frac{(1 - \cos (x y)) \tan (x y)}{y^{3}}$ . Then the number of solutions of the equation f(x) = sin x, $x \in R$ is: [2026]

Smart Achievers · Accepted Answer

(3)

Given, $f (x) = \lim_{y \to 0} \frac{(1 - \cos (x y)) \tan (x y)}{y^{3}}$

$= \lim_{y \to 0} x^{2} \frac{(1 - \cos (x y))}{{(x y)}^{2}} \times \frac{\tan (x y)}{(x y)} \times x$

$\Rightarrow f (x) = \frac{x^{3}}{2}$

Now for solution f(x) = sin x

$∴$ Number of solutions = 3

Solution

Q.
Let $f (x) = \lim_{y \to 0} \frac{(1 - \cos (x y)) \tan (x y)}{y^{3}}$ . Then the number of solutions of the equation f(x) = sin x, $x \in R$ is: [2026]

1 0

2 2

3 3

4 1

Ans.
(3)

Given, $f (x) = \lim_{y \to 0} \frac{(1 - \cos (x y)) \tan (x y)}{y^{3}}$

$= \lim_{y \to 0} x^{2} \frac{(1 - \cos (x y))}{{(x y)}^{2}} \times \frac{\tan (x y)}{(x y)} \times x$

$\Rightarrow f (x) = \frac{x^{3}}{2}$

Now for solution f(x) = sin x

$∴$ Number of solutions = 3

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Solution

Q. Let f(x)=limy→0(1–cos(xy))tan(xy)y3. Then the number of solutions of the equation f(x) = sin x, x∈R is: [2026]

1 0

2 2

3 3

4 1

Ans. (3) Given, f(x)=limy→0(1–cos(xy))tan(xy)y3 =limy→0x2(1–cos(xy))(xy)2×tan(xy)(xy)×x ⇒ f(x)=x32 Now for solution f(x) = sin x ∴ Number of solutions = 3

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Q.
Let $f (x) = \lim_{y \to 0} \frac{(1 - \cos (x y)) \tan (x y)}{y^{3}}$ . Then the number of solutions of the equation f(x) = sin x, $x \in R$ is: [2026]

Ans.
(3)

Given, $f (x) = \lim_{y \to 0} \frac{(1 - \cos (x y)) \tan (x y)}{y^{3}}$

$= \lim_{y \to 0} x^{2} \frac{(1 - \cos (x y))}{{(x y)}^{2}} \times \frac{\tan (x y)}{(x y)} \times x$

$\Rightarrow f (x) = \frac{x^{3}}{2}$

Now for solution f(x) = sin x

$∴$ Number of solutions = 3