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How do you find the limit of # (sin^2(x^2))/(x^4)# as x approaches 0?

Answer 1

#1#

Let #f(x)=(sin^2(x^2))/x^4#

#implies f'(x)=lim_(x to 0) (sin^2(x^2))/x^4#

#implies f'(x)=lim_(x to 0) (sin(x^2)*sin(x^2))/x^4=lim_(x to 0) {sin(x^2)/x^2*sin(x^2)/x^2}=lim_(x to 0)sin(x^2)/x^2lim_(x to 0)sin(x^2)/x^2*=1*1=1#

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Answer 2

Christopher Allers

To find the limit of (sin^2(x^2))/(x^4) as x approaches 0, we can use the limit properties and trigonometric identities. By applying the limit properties, we can simplify the expression. Since sin(x)/x approaches 1 as x approaches 0, we can rewrite the expression as (sin(x^2)/x^2)^2 / x^2. Simplifying further, we get (sin(x^2))^2 / (x^2)^3. As x approaches 0, sin(x^2) also approaches 0, so the numerator becomes 0. The denominator, (x^2)^3, becomes 0 as well. Therefore, the limit of (sin^2(x^2))/(x^4) as x approaches 0 is 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.