# Limit of (arcsen(2x)-2arcsen(x))/x^3 as x->0

• tsuwal
In summary, the limit of (arcsen(2x)-2arcsen(x))/x^3 as x->0 is an 0/0 indeterminacy and can be solved using L'Hopital's rule. The process involves simplifying the expression by multiplying it by the conjugate root and using the derivative of the trigonometric function.
tsuwal

## Homework Statement

Limit of (arcsen(2x)-2arcsen(x))/x^3 as x->0

## Homework Equations

arsen(x)'=1/Sqrt(1-x^2)

## The Attempt at a Solution

It's an 0/0 indeterminancy, to solve it, I had to use L'hopital's rule once simplify de expression, multiplying by the conjugate root. Is there an easier way?

As you don't get rid of the trigonometric functions without l'Hospital, I think this ok.

$$\frac{\frac{2}{\sqrt{1-(2x)^2}}-\frac{2}{\sqrt{1-(x)^2}}}{x^2} = 2 \frac{1}{\frac{1}{\sqrt{1-(2x)^2}}+\frac{1}{\sqrt{1-(x)^2}}} \frac{\frac{1}{1-(2x)^2}-\frac{1}{1-(x)^2}}{x^2} \to \frac{2}{2} \frac{\frac{1}{1-(2x)^2}-\frac{1}{1-(x)^2}}{x^2} \to \dots$$

Thanks!

## 1. What is the definition of a limit in calculus?

The limit of a function is the value that a function approaches as the input approaches a certain value or approaches a point on the function. It is used to describe the behavior of a function at a specific point or as the input approaches a particular value.

## 2. What is the limit of a function as x approaches 0?

The limit of a function as x approaches 0 is the value that the function approaches as the input, x, gets closer and closer to 0. This can be found by evaluating the function at values of x that are very close to 0, either from the left or right side of 0.

## 3. How do you find the limit of a function algebraically?

To find the limit of a function algebraically, you can use various techniques such as factoring, simplifying, or using algebraic manipulations to rewrite the function. Then, you can substitute the given value for x and evaluate the function. If the resulting value is undefined, you may need to use L'Hopital's Rule or other methods to find the limit.

## 4. What is the limit of (arcsen(2x)-2arcsen(x))/x^3 as x approaches 0?

The limit of (arcsen(2x)-2arcsen(x))/x^3 as x approaches 0 is 1/6. This can be found by using algebraic manipulations and the limit definition of the derivative. Alternatively, you can use the power series expansion of arcsen(x) to evaluate the limit.

## 5. Why is the limit of (arcsen(2x)-2arcsen(x))/x^3 as x approaches 0 important?

The limit of (arcsen(2x)-2arcsen(x))/x^3 as x approaches 0 is important because it helps us understand the behavior of the function near the point where x=0. This limit can also be used to find the derivative of the function at x=0, which is useful in many applications of calculus.

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