L’Hôpital’s Rule
Find the limits in Exercises 103–110.

Question

L’Hôpital’s Rule

Find the limits in Exercises 103–110.

105. lim(x→∞) x arctan(2/x)

Accepted Answer

Hello. In this video, we are going to be evaluating the following limit by using the Hopitos rule. So we are told, or we are given the limit, the limit as x approaches infinity of x multiplied by arc sine, which is equivalent to sine inverse of 3 divided by x. Now, in order to determine if we need to use the Holpito's rule, we need to check to see if our function is an indeterminate form. So by plugging in the limit, we get infinity multiplied by sine inverse of 3 divided by infinity, which is equivalent to 0. So this simplifies to infinity multiplied by 0, which is an indeterminate form. And so what we need to do is we need to rewrite the given limit in order to apply Lahopito's rule, since Holohopito's rule only works on rational functions. Well, here we can go ahead and rewrite x as a rational function of sin inverse divided by 1 divided by x, so we can rewrite the given limit as the limit as x approaches infinity of sin inverse of 3 divided by x, all divided by 1 divided by x. Now Lohopito's rule asks us to take the derivative of both the numerator and denominator of our rational function. So by applying the Hopito's rule, we will need to take the derivative of sin inverse of 3 divided by X and 1 divided by X. Now recall that the derivative of sin inverse of U is equal to U divided by the square root of 1 minus U2. In this case here, U is equal to 3 divided by X, and that means that U is equal to -3 divided by X2. And so by taking the derivative of the sine inverse function, our numerator is going to be 3 divided by x 2, all divided by the square root of 1. -3 3 divided by X quantity 2, which is 9 divided by X2. So this is going to be the numerator of our function by using the Hopito's rule. And the derivative of our denominator 1 divided by x is going to be -1 divided by x 2. Now we can further simplify this function. By multiplying both the numerator and denominator by the reciprocal of our denominator, we can simplify the complex fraction to be the limit as x approaches infinity of the quantity positive 3 divided by the square root of 1 minus 9 divided by x squared. Now, if we were to apply the limit, 9 divided by infinity is going to go to 0, so that will cancel out to be a zero quantity. And what we are left with is 3 divided by the square root of 1, which further simplifies to be 3. This is going to be the value of the given limit, and the overall solution to this problem is going to be b. So I hope this video helps you in understanding how to use the Hopitos rule, and we will go ahead and see you all in the next video.

L’Hôpital’s RuleFind the limits in Exercises 103–110.105. lim(x→∞) x arctan(2/x)

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L’Hôpital’s Rule
Find the limits in Exercises 103–110.
105. lim(x→∞) x arctan(2/x)