Mathematics: Finding the Limit of (1 – cos(x))/x as x Approaches 0 Using Trigonometric Techniques

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

To find the limit of the expression (1 – cos(x))/x as x approaches 0, we can use a common trigonometric limit

To find the limit of the expression (1 – cos(x))/x as x approaches 0, we can use a common trigonometric limit.

Step 1: Simplify the expression:
(1 – cos(x))/x = 1/x – cos(x)/x

Step 2: Apply the limit to each term separately:
lim(x -> 0) (1/x) – lim(x -> 0) (cos(x)/x)

Step 3: Evaluate each limit:
The limit of 1/x as x approaches 0 is infinity (∞) because as x gets closer to 0, the value of 1/x becomes larger and larger.

The limit of cos(x)/x as x approaches 0 is an indeterminate form, as the numerator and denominator both approach 0. We need to use L’Hopital’s Rule to evaluate this limit.

Applying L’Hopital’s Rule:
Differentiate the numerator and denominator separately:
d/dx (cos(x)) = -sin(x)
d/dx (x) = 1

Now we have:
lim(x -> 0) (cos(x)/x) = lim(x -> 0) (-sin(x)/1)

Step 4: Evaluate the new limit:
The limit of -sin(x)/1 as x approaches 0 is -sin(0)/1 = 0/1 = 0.

Step 5: Combine the limits from step 2:
lim(x -> 0) (1/x) – lim(x -> 0) (cos(x)/x) = ∞ – 0 = ∞

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

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