Applying L’Hôpital’s Rule to Find the Limit of (1-cos(x))/x as x approaches 0

lim x->0 1-cosx/x

To find the limit of the expression (1-cos(x))/x as x approaches 0, we can apply the concept of L’Hôpital’s rule

To find the limit of the expression (1-cos(x))/x as x approaches 0, we can apply the concept of L’Hôpital’s rule. L’Hôpital’s rule states that if we have an indeterminate form of the type 0/0 or ∞/∞ when finding a limit, we can differentiate the numerator and denominator separately until we get a result that is no longer indeterminate.

Let’s apply L’Hôpital’s rule to this problem:

lim x->0 (1-cos(x))/x

Since both the numerator (1-cos(x)) and the denominator (x) go to 0 as x approaches 0, we have an indeterminate form of 0/0.

Now, let’s differentiate the numerator and denominator separately:

lim x->0 sin(x)/1

Differentiating the numerator, we get cos(x). Differentiating the denominator, we get 1.

So, the limit becomes:

lim x->0 cos(x)/1

Now, we can substitute x = 0 into the expression:

cos(0)/1 = 1/1 = 1

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

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