Limits and Infinity

Find the...

Question

Limits and Infinity

Find the limits in Exercises 37–46.

x²/³ + x⁻¹

lim --------------------

x→∞ x²/³ + cos²x

Accepted Answer

Welcome back everyone. Determine the limit of the function F of X equals 4x the power of 1/4 minus 5 X to the power of -3, divided by X to the power of 1/4 plus 3 2 X X approaches infinity. We are given 4 answer choices A says 14th, B4, C 0, and D does not exist. So what we're going to do is simply write down the limit, limit as x approaches infinity. Our function is 4 x to the power of 1/4 minus 5x the power of -3. So that's basically 5 divided by X cubed, right? We can just rewrite it in this form using the reciprocal rule, and then in the denominator we have X to the power of 1/4 + 3 square of X. So what we want to do is simply analyze a possible result. If X approaches infinity, we have 4 X of the power of 1/4, so this term is going to approach infinity. 5 divided by X cubed or 5 divided by infinity is going to approach 0, so we just have infinity in the numerator for the denominator. X the power of 1/4 S X approaches infinity also approaches infinity, and we're adding sine squared, which basically oscillates between 0 and 1 inclusive, right, so it has no effect on infinity. And when we have this form and X approaches infinity, one of the ways to solve for the limit is to divide our numerator and the denominator of the irrational fraction. Or a rational function that we more accurate to say by x with the highest exponent in the denominator and that's x to the power of 14th, right? So we're going to divide all of our terms by x to the power of 1/4, which gives us 4 in the numerator. Minus now if we divide 5 x to the power of -3 by x the power of 1/4, we can use the properties of exponents, and that is going to be 5 x to the power of -3 minus 1/4. And now if we find the common denominator, that's just 5 X to the power of negative 13 divided by 4, or basically 5 divided by X to the power of 13 divided by 4. So this is how we get our next term. It's 5 divided. 5 x actually if we use the reciprocal rule, yes, we can write it as 5 divided by X to the power of 13 divided by 4. And then for our denominator, if we divide all terms by X to the power of 1/4, we're going to get 1 + 3 squared of x divided by. X to the power of 1/4 and now we have to understand that as X approaches infinity, we have 55 divided by infinity which approaches 0. Whenever we have a constant divided by an infinitely large number, we get a limit of 0. And now for our sine squared, we have to understand that it's oscillating between 0 and 1. When we multiply by 3, it's basically oscillating between 0 and 3, so we're changing the amplitude, right? But basically it is oscillating in a fixed range while x approaches infinity. So this means that the limit is also 0 because we have a number between 0 and 3 divided by an infinitely large number, right? And from here we can see that we're only left with two terms. One of them is 4 and the other one is 1. So 4 divided by 1 is a finite number which is 4. And this is our limit for this problem which corresponds to the answer choice B. Thank you for watching.

Limits and Infinity

Find the limits in Exercises 37–46.

x²/³ + x⁻¹
lim --------------------
x→∞ x²/³ + cos²x

Key Concepts

Limits

Dominant Terms

Trigonometric Functions and Limits

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