Understanding the Behavior and Limit of sin(x)/x as x Approaches 0 in Mathematics

limit as x approaches 0: sinx/x

To find the limit of sin(x)/x as x approaches 0, we can analyze the behavior of the function as x gets closer and closer to 0

To find the limit of sin(x)/x as x approaches 0, we can analyze the behavior of the function as x gets closer and closer to 0.

Let’s start by evaluating sin(x)/x for some values of x near 0.

When x = 0.1: sin(0.1)/0.1 = 0.99833…
When x = 0.01: sin(0.01)/0.01 = 0.999983…
When x = 0.001: sin(0.001)/0.001 = 0.99999983…

Notice that as x gets smaller, the value of sin(x)/x gets very close to 1.

Now, let’s examine the limit formally using algebraic manipulation.

We know that sin(0) = 0, so the expression sin(x)/x is undefined at x = 0 due to division by 0.

However, we can use the properties of limits to determine the limit of sin(x)/x as x approaches 0. We can rewrite sin(x)/x as sin(x)/x * x/x = (sin(x) * x) / (x^2).

Dividing both the numerator and the denominator by x, we get (sin(x) * x) / (x^2) = sin(x) / x, where x ≠ 0.

Now, we can take the limit as x approaches 0 of sin(x)/x: lim x→0 (sin(x) / x).

Applying the limit rule for products, we have:

lim x→0 (sin(x) * x) / (x^2) = (lim x→0 sin(x)) * (lim x→0 x) / (lim x→0 (x^2)).

Since lim x→0 sin(x) = sin(0) = 0, lim x→0 x = 0, and lim x→0 (x^2) = 0, we have:

0 * 0 / 0 = 0/0

At this point, we have an indeterminate form, which means we cannot directly determine the limit.

To resolve this indeterminate form, we can apply L’Hôpital’s rule, which states that if we have a limit of the form 0/0 or ∞/∞, we can take the derivative of the numerator and the denominator until we obtain a limit that can be directly evaluated.

Let’s differentiate both the numerator and the denominator with respect to x:

d/dx (sin(x)) = cos(x)
d/dx (x) = 1
d/dx (x^2) = 2x

Now, let’s reevaluate the limit:

lim x→0 (sin(x) * x) / (x^2) = [d/dx (sin(x))] / [d/dx (x^2)]
= cos(x) / 2x

Now, we can substitute x = 0 into the expression cos(x) / 2x:

lim x→0 (cos(x) / 2x)
= (cos(0) / 2(0))
= (1 / 0)

Now we have another indeterminate form, 1/0, which is undefined.

Therefore, the limit of sin(x)/x as x approaches 0 does not exist.

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