use L'Hospital's rule to evaluate: limit as x approaches to 0 of (1/sinx -1/x)

To evaluate the limit using L'Hospital's Rule, follow these steps:

Step 1: Write the expression in the form of an indeterminate form (0/0 or ∞/∞) by simplifying it as much as possible. In this case, we have (1/sin(x) - 1/x).

Step 2: Differentiate the numerator and the denominator separately.

For the numerator:
The derivative of 1/sin(x) is (sin(x))' = cos(x).
For the denominator:
The derivative of 1/x is (1/x)' = -1/x^2.

Step 3: Rewrite the expression using the derivatives obtained in step 2. The expression becomes (cos(x) - 1/x^2).

Step 4: Now, evaluate the limit as x approaches 0 using the newly obtained expression. The limit should be straightforward to calculate since there are no indeterminate forms present anymore.

lim(x->0) of (cos(x) - 1/x^2) = cos(0) - 1/0^2 = 1 - 1/0 = undefined.

Note: The limit of the original expression as x approaches 0 using L'Hospital's Rule is undefined.

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