Find the limits in Exercises 59–62. Write ∞ or −∞ where approp...

Question

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as

a. x → 0⁺

Accepted Answer

Welcome back everyone. Find the limit as x approaches 0 from the right for the function H X equals 5 divided by X to the power of 2/3 plus 7 divided by X + 1 of 4/3. A says infinity, B negative infinity, C 0, and D7. Let's write down the limit. Limit as X approaches 0 from the right of H of X, which is 5 divided by. Cub root of X2. Now we can just rewrite x the power of 2/3 as cube root of X2 using the properties of exponents. Additionally, because we have a sum, we can distribute the limits, so we're basically adding the limit as x approaches 0 from the right. Of 7 divided by now applying the properties of exponents we can see that we have 4/3, so this means that we're taking cube root of X + 1 raise the power of 4. And now let's analyze each limit. For the first limit, if we perform direct substitution, this gives us division by 0. A non-zero number divided by 0 leads to. The limit of infinity. We basically have to determine the sign. So 5 is a positive number and now because we're approaching 0 from the right, this means that X is greater than 0, and cube root of a positive number, which is also squared, leads to the positive number, right? This means that our denominator is going to be positive. As a result, the ratio of 2 positive numbers is going to be positive, so we get positive infinity. It is really important to understand the sign of infinity in this context. Now analyzing the second limit, we can actually perform direct substitution which gives us 7 divided by cube root. Of 0 + 1 raises the power of 4. So once the power of 4 is 1. The ratio becomes 7 and now we have addition. So we're adding infinity and 7, right? So if we're adding 7 and infinity, well, this basically leads to infinity. It doesn't change the fact that the first limit grows in the first function grows infinitely large as X approaches 0 from the right. So adding a constant to infinity just leads to the same infinity, and therefore, the answer to this problem corresponds to the answer choice A, infinity. Thank you for watching.

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as

a. x → 0⁺

Key Concepts

Limit of a Function

One-Sided Limits

Indeterminate Forms

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