Exploring the Limit of (1 – cos(x))/x as x Approaches 0 using L’Hôpital’s Rule and Trigonometric Identities

lim x->0 1-cosx/x

To find the limit of the function (1 – cos(x))/x as x approaches 0, we can use L’Hôpital’s Rule or some trigonometric identities

To find the limit of the function (1 – cos(x))/x as x approaches 0, we can use L’Hôpital’s Rule or some trigonometric identities.

Method 1: L’Hôpital’s Rule
L’Hôpital’s Rule is applicable when we have an indeterminate form of 0/0. In this case, the limit of (1 – cos(x))/x as x approaches 0 is in the form of 0/0, so we can apply this rule.

Taking the derivative of the numerator and denominator separately, we have:

lim x->0 [(d/dx)(1 – cos(x))]/(d/dx)x

Differentiating 1 with respect to x gives us 0 because it is a constant. For the derivative of cos(x), we use the chain rule and get -sin(x). The derivative of x with respect to x is simply 1.

So, the expression becomes:

lim x->0 [-sin(x)]/1

Now, we can evaluate the limit as x approaches 0:

lim x->0 -sin(x) = -sin(0) = 0

Therefore, the limit of (1 – cos(x))/x as x approaches 0 is 0.

Method 2: Trigonometric Identity
We can also use a trigonometric identity to simplify the expression before evaluating the limit.

The identity we will use is: cos(x) = 1 – 2sin^2(x/2)

Let’s substitute this identity into the function:

(1 – cos(x))/x = (1 – (1 – 2sin^2(x/2)))/x
= 2sin^2(x/2)/x

Now, we have a new expression. Let’s proceed with evaluating the limit using this new expression.

lim x->0 2sin^2(x/2)/x

Next, we can apply the double angle formula for sine: sin^2(x/2) = (1 – cos(x))/2

Substituting this into the expression, we get:

lim x->0 2((1 – cos(x))/2)/x
= lim x->0 (1 – cos(x))/x

Now, we have the same expression we started with. We already know from Method 1 that this limit is equal to 0.

Therefore, regardless of the method used, the limit of (1 – cos(x))/x as x approaches 0 is 0.

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