Evaluate the Limit

Problem

(lim_x→0)(1/sin(x)−1/x)

Solution

1. Combine the fractions by finding a common denominator to transform the expression into a single quotient.

1/sin(x)−1/x=(x−sin(x))/(x*sin(x))

2. Identify the indeterminate form by evaluating the limit as x→0

(lim_x→0)((x−sin(x))/(x*sin(x)))=(0−0)/(0⋅0)=0/0

3. Apply L'Hôpital's Rule by differentiating the numerator and the denominator with respect to x

(lim_x→0)(d(x−sin(x))/d(x)/d(x*sin(x))/d(x))=(lim_x→0)((1−cos(x))/(sin(x)+x*cos(x)))

4. Check the indeterminate form again for the new expression.

(1−cos(0))/(sin(0)+0⋅cos(0))=(1−1)/(0+0)=0/0

5. Apply L'Hôpital's Rule a second time to resolve the 0/0 form.

(lim_x→0)(d(1−cos(x))/d(x)/d(sin(x)+x*cos(x))/d(x))=(lim_x→0)(sin(x)/(cos(x)+cos(x)−x*sin(x)))

6. Simplify the denominator before evaluating the limit.

(lim_x→0)(sin(x)/(2*cos(x)−x*sin(x)))

7. Substitute the value x=0 into the simplified expression.

sin(0)/(2*cos(0)−0⋅sin(0))=0/(2*(1)−0)=0/2=0

Final Answer

(lim_x→0)(1/sin(x)−1/x)=0

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