Finding the Limit of (1 – cos(x)) / x as x Approaches 0 | Simplification and L’Hôpital’s Rule

lim x->0 1-cosx/x =

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

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

Step 1: Simplify the expression
We can rewrite the numerator as 2sin^2(x/2) using the identity 1 – cos(x) = 2sin^2(x/2).
So, the expression becomes (2sin^2(x/2)) / x.

Step 2: Apply L’Hôpital’s rule
L’Hôpital’s rule states that if we have a limit of the form 0/0 or ∞/∞, we can take the derivative of the numerator and the derivative of the denominator separately, and then assess the limit again.

Taking the derivative of 2sin^2(x/2), we get: d/dx [2sin^2(x/2)] = 2sin(x/2) * cos(x/2) * (1/2).

Taking the derivative of x, we get: d/dx [x] = 1.

Now let’s evaluate the limit using L’Hôpital’s rule:
lim x->0 (2sin(x/2) * cos(x/2) * (1/2)) / 1
= lim x->0 (sin(x/2) * cos(x/2)) / 2

Step 3: Evaluate the limit
Now, we can evaluate the limit. As x approaches 0, sin(x/2) approaches 0 and cos(x/2) approaches 1.
Therefore, the limit becomes:
lim x->0 (0 * 1) / 2
= 0.

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

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