Evaluate the Limit

Problem

(lim_t→0)((t*(1−cos(t)))/(t−sin(t)))

Solution

1. Identify the indeterminate form by substituting t=0 into the expression, which results in (0*(1−1))/(0−0)=0/0

2. Apply L'Hôpital's Rule by differentiating the numerator and the denominator with respect to t

3. Differentiate the numerator using the product rule: (d(t)*(1−cos(t)))/d(t)=(1)*(1−cos(t))+(t)*(sin(t))

4. Differentiate the denominator: d(t−sin(t))/d(t)=1−cos(t)

5. Rewrite the limit as (lim_t→0)((1−cos(t)+t*sin(t))/(1−cos(t)))

6. Identify the indeterminate form again as t→0 which is (1−1+0)/(1−1)=0/0 requiring a second application of L'Hôpital's Rule.

7. Differentiate the new numerator: d(1−cos(t)+t*sin(t))/d(t)=sin(t)+sin(t)+t*cos(t)=2*sin(t)+t*cos(t)

8. Differentiate the new denominator: d(1−cos(t))/d(t)=sin(t)

9. Rewrite the limit as (lim_t→0)((2*sin(t)+t*cos(t))/sin(t))

10. Simplify the expression by dividing both terms in the numerator by sin(t) resulting in (lim_t→0)(2+(t*cos(t))/sin(t))

11. Evaluate the remaining limit using the standard limit (lim_t→0)(sin(t)/t)=1 which implies (lim_t→0)(t/sin(t))=1

12. Substitute the values: 2+(1)*(cos(0))=2+1=3

Final Answer

(lim_t→0)((t*(1−cos(t)))/(t−sin(t)))=3

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