Evaluate the Limit limit as x approaches 0 of csc(x)-cot(x)

Problem

(lim_x→0)(csc(x)−cot(x))

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine to identify the indeterminate form.

csc(x)−cot(x)=1/sin(x)−cos(x)/sin(x)

2. Combine the terms into a single fraction with a common denominator.

(1−cos(x))/sin(x)

3. Evaluate the limit at x=0 to check the form.

(1−cos(0))/sin(0)=(1−1)/0=0/0

4. Apply L'Hôpital's Rule by differentiating the numerator and the denominator separately.

d(1−cos(x))/d(x)=sin(x)

d(sin(x))/d(x)=cos(x)

5. Calculate the new limit as x approaches 0

(lim_x→0)(sin(x)/cos(x))=sin(0)/cos(0)

0/1=0

Final Answer

(lim_x→0)(csc(x)−cot(x))=0

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