lim x→0 sin2x/x formula | lim x→∞ sin2x/x formula

The limit of sin2x/x when x→0 is equal to 2 and the limit of sin2x/x when x→∞ is equal to 0. Here we find the limits of sin2x/x when x tends to 0 or ∞.

The formulas of the limits of sin2x/x when x→0 or x→∞ is given below:

1. lim_x→0 sin2x/x = 2.
2. lim_x→∞ sin2x/x = 0.

Find lim_x→0 sin2x/x

Answer: The function sin2x/x has the limit 2 when x tends to 0.

Explanation:

We have

lim_x→0 sin2x/x

= lim_x→0 (sin2x/2x × 2)

= 2 lim_x→0 sin2x/2x

Let z=2x, so that z→0 when x→0. Then the above limit will be

= 2 lim_z→0 sinz/z

= 2 × 1 as we know that lim_x→0 sinx/x = 1.

= 2

So the value of lim x→0 sin2x/x is equal to 2.

Find lim_x→∞ sin2x/x

Answer: The function sin2x/x has the limit 0 when x tends to ∞.

Explanation:

As sine function takes values in [-1, 1], we have that -1 ≤ sin 2x ≤ 1. As x approaches to ∞, dividing by x it follows that

$-\dfrac{1}{x} \leq \dfrac{\sin 2x}{x} \leq \dfrac{1}{x}$

Now, taking the limit x→∞ we obtain that

$- \lim\limits_{x \to \infty} \dfrac{1}{x}$ $\leq \lim\limits_{x \to \infty} \dfrac{\sin 2x}{x}$ $\leq \lim\limits_{x \to \infty} \dfrac{1}{x}$

⇒ 0 ≤ lim_x→∞ $\dfrac{\sin 2x}{x}$ ≤ 0

Thus, by the squeeze theorem of limits, the limit of sin2x/x when x approaches to infinity is equal to 0.

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