lim_(x->0) (1/x^2-1/(sin^2(x)))
To find the limit of the given function as x approaches 0, follow these steps:
Step 1: Simplify the expression:
Start by simplifying the denominator. We can rewrite the expression as follows:
1/x^2 - 1/(sin^2(x)) = (sin^2(x) - x^2) / (x^2 * sin^2(x))
Step 2: Factor out the numerator:
(sin^2(x) - x^2) can be factored into (sin(x) + x)(sin(x) - x).
So, now we have:
[(sin(x) + x)(sin(x) - x)] / (x^2 * sin^2(x))
Step 3: Cancel out common factors:
We can cancel out the common factor of (sin(x) - x) in the numerator and denominator:
(sin(x) + x) / (x^2 * sin^2(x))
Step 4: Evaluate the limit:
Now, as x approaches 0, we can substitute x = 0 into the expression:
(sin(0) + 0) / (0^2 * sin^2(0)) = 0 / (0 * 0)
The numerator is 0, and the denominator is also 0, which is an indeterminate form.
Therefore, to evaluate this limit more accurately, we can apply L'Hopital's Rule.
Step 5: Apply L'Hopital's Rule:
Differentiate the numerator and denominator with respect to x:
The derivative of sin(x) + x with respect to x is cos(x) + 1.
The derivative of x^2 * sin^2(x) with respect to x can be found using the product rule:
(differentiate x^2) * sin^2(x) + x^2 * (differentiate sin^2(x))
Which simplifies to:
2x * sin^2(x) + 2x^2 * sin(x) * cos(x)
Now we can calculate the limit again using the differentiated expressions:
lim_(x->0) [(cos(x) + 1) / (2x * sin^2(x) + 2x^2 * sin(x) * cos(x))]
At this point, we can substitute x = 0 into the expression:
[(cos(0) + 1) / (2 * 0 * sin^2(0) + 2 * 0^2 * sin(0) * cos(0))]
The numerator is 2, and the denominator is 0.
Therefore, the limit of the given function as x approaches 0 is undefined since we obtained an indeterminate form of 2/0.