Understanding L’Hôpital’s Rule | Evaluating the Limit of sin(x)/x as x Approaches 0

limit as x approaches 0: sinx/x

To find the limit as x approaches 0 of the expression sin(x)/x, we can use the concept of L’Hôpital’s Rule

To find the limit as x approaches 0 of the expression sin(x)/x, we can use the concept of L’Hôpital’s Rule.

L’Hôpital’s Rule states that if we have a limit in the form 0/0 or ∞/∞, and if the limit of the derivative of the numerator divided by the derivative of the denominator exists, then the limit of the original expression exists as well and is equal to the limit of the derivative quotient.

Let’s apply L’Hôpital’s Rule to the expression sin(x)/x:

We take the derivative of the numerator and denominator separately. The derivative of sin(x) is cos(x), and the derivative of x is 1. So the derivative quotient is:

(cos(x))/1

Now, let’s find the limit as x approaches 0 of this derivative quotient:

lim(x → 0) [cos(x)/1]

As x approaches 0, cos(x) approaches 1. So the limit is:

lim(x → 0) [1/1] = 1

Therefore, the limit as x approaches 0 of sin(x)/x is equal to 1.

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