Evaluating the Limit of sin(x) / x as x Approaches 0: L’Hôpital’s Rule and Trigonometric Identities

lim x->0 sinx/x

To evaluate the limit of sin(x) / x as x approaches 0, we can use L’Hôpital’s rule or some trigonometric identities

To evaluate the limit of sin(x) / x as x approaches 0, we can use L’Hôpital’s rule or some trigonometric identities.

Using L’Hôpital’s rule:
We start by taking the derivative of both the numerator and the denominator with respect to x.
The derivative of sin(x) is cos(x) and the derivative of x is 1.
Therefore, the limit can be rewritten as:
lim x->0 cos(x) / 1

Since we have a straightforward function in the numerator and denominator, we can evaluate the limit directly.
As x approaches 0, cos(x) approaches 1.
Thus, the limit simplifies to:
lim x->0 1 / 1 = 1

Using trigonometric identities:
We can also rewrite sin(x) / x using a trigonometric identity.
The identity is:
sin(x) = x * (1 / x)

Substituting this identity into the limit expression, we get:
lim x->0 x * (1 / x) / x

Simplifying the expression, we have:
lim x->0 1 / x

As x approaches 0, 1 / x will approach either positive or negative infinity, depending on which side of 0 we are approaching from.
Therefore, the limit of sin(x) / x as x approaches 0 is undefined.

Overall, the limit of sin(x) / x as x approaches 0 can be evaluated using either L’Hôpital’s rule (1) or the trigonometric identity (undefined).

Note: It is important to note that limits are concepts used in mathematics to analyze the behavior of functions as the input approaches a certain value, and they may not always result in a finite value.

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