Finding the Limit as x Approaches 0 of (1 – cosx)/x: L’Hopital’s Rule vs Trigonometric Identity

limit as x approaches 0: 1-cosx/x

To find the limit as x approaches 0 of the expression (1 – cosx)/x, we can use L’Hopital’s rule or apply a trigonometric identity

To find the limit as x approaches 0 of the expression (1 – cosx)/x, we can use L’Hopital’s rule or apply a trigonometric identity.

Method 1: L’Hopital’s Rule
1. Step: Find the derivative of the numerator and denominator.
The derivative of 1 – cosx with respect to x is sinx.
The derivative of x with respect to x is 1.

2. Step: Take the limit of the derivatives as x approaches 0.
lim (x approaches 0) sinx/1 = sin0/1 = 0/1 = 0.

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

Method 2: Trigonometric Identity
1. Step: Rewrite the expression using a trigonometric identity.
(1 – cosx)/x can be rewritten as (1 – cosx)(1 + cosx)/x(1 + cosx).

2. Step: Simplify the expression using the trigonometric identity cos^2(x) = 1 – sin^2(x).
(1^2 – cos^2(x))/(x(1 + cosx)).
sin^2(x) simplifies to 1 – cos^2(x), so we have (1^2 – (1 – sin^2(x)))/(x(1 + cosx)).
This further simplifies to (1 – 1 + sin^2(x))/(x(1 + cosx)).
sin^2(x) simplifies to 0 + sin^2(x), so we have (sin^2(x))/(x(1 + cosx)).

3. Step: Evaluate the limit as x approaches 0 by substituting the value.
lim (x approaches 0) (sin^2(x))/(x(1 + cosx)).
When we substitute x = 0, sin^2(x) becomes sin^2(0) which is equal to 0.
We get 0/(0(1 + cos0)) = 0/(0(1 + 1)) = 0/0.

Since we have an indeterminate form (0/0), we can apply L’Hopital’s rule again.

4. Step: Differentiate the numerator and denominator.
The derivative of sin^2(x) with respect to x is 2sin(x)cos(x).
The derivative of x(1 + cosx) with respect to x is 1 + cosx – xsinx.

5. Step: Evaluate the limit of the derivatives as x approaches 0.
lim (x approaches 0) 2sin(x)cos(x)/(1 + cosx – xsinx).
When we substitute x = 0, sin(x) becomes sin(0) which is equal to 0, and cos(x) becomes cos(0) which is equal to 1.
We get 2(0)(1)/(1 + 1 – 0) = 0/2 = 0.

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

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