Determining the Limit of (cos(x) – 1)/x as x Approaches 0

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

To evaluate the limit of the expression lim x->0 (cos(x) – 1)/x, we can use algebraic manipulations along with the knowledge of some trigonometric limits

To evaluate the limit of the expression lim x->0 (cos(x) – 1)/x, we can use algebraic manipulations along with the knowledge of some trigonometric limits. Here’s how we can approach it step by step:

Step 1: Simplify the expression.
Using trigonometric identity cos(x) – 1 = -2sin^2(x/2), we can rewrite the expression as:
lim x->0 (-2sin^2(x/2))/x

Step 2: Further simplify the expression.
Since sin(x) can be expressed as sin(x/2) * cos(x/2), we can rewrite the expression as:
lim x->0 (-2(sin(x/2))^2)/x

Step 3: Apply a limit property.
The limit property states that if the limit of f(x)/g(x) exists as x approaches a, and g(x) is nonzero at a, then the limit of f(x)/g(x) also equals the limit of f(x) divided by the limit of g(x) as x approaches a. In this case, both sin(x/2) and (sin(x/2))^2 have a limit of 0 as x approaches 0. Therefore, we can apply the limit property and rewrite the expression as:
-2 * (lim x->0 (sin(x/2))^2) / (lim x->0 x)

Step 4: Evaluate the limits separately.
The two limits in the expression are:
lim x->0 (sin(x/2))^2
lim x->0 x

The limit of (sin(x/2))^2 can be calculated as follows:
lim x->0 (sin(x/2))^2 = sin^2(0/2) = sin^2(0) = 0

The limit of x as x approaches 0 is simply 0.

Step 5: Calculate the final limit.
Now that we have the limits of the separate components, we can calculate the final limit by substituting the results back into the expression:
lim x->0 (-2(sin(x/2))^2)/x = -2 * 0 / 0

However, we reached an indeterminate form of 0/0, which means we cannot determine the limit immediately. In such cases, we need to use a different method to evaluate the limit.

Step 6: Apply L’Hôpital’s rule.
L’Hôpital’s rule states that if the limit of f(x)/g(x) as x approaches a is of the form 0/0 or ∞/∞, then the limit of f(x)/g(x) is equal to the limit of the derivative of f(x) divided by the derivative of g(x) as x approaches a.

In our case, since we have an indeterminate form of 0/0, we can apply L’Hôpital’s rule. Taking the derivative of the numerator and denominator separately gives us:
lim x->0 [(-4sin(x/2) * (1/2)cos(x/2))]/1

Simplifying this yields:
lim x->0 [-2sin(x/2)cos(x/2)]/1
lim x->0 -2sin(x/2)cos(x/2)

Step 7: Calculate the limit again.
Now we have the new expression after applying L’Hôpital’s rule. We can evaluate it by plugging in x = 0 directly:
lim x->0 -2sin(x/2)cos(x/2) = -2sin(0/2)cos(0/2) = -2sin^2(0)cos(0) = 0

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

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