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 / x²/³ + 2 / (x − 1)²/³ ) as

a. x → 0⁺

Accepted Answer

Welcome back everyone. Find the limit as x approaches 0 from the right for the function H of X equals 2 divided by x rates to the power of 1/3 + 3 divided by X + 8 raise to the power of 1/3. We're given 4 answer choices A says 3 halves, B 0, C infinity, and D negative infinity. Let's write down the limit. Limits X approaches 0 from the right. Of 2 divided by X raises the power of 1/3 + 3 divided by. X + 8 raises the power of 1/3. If we use direct substitution, we will notice that the first term leads to 2 divided by a 0, which is undefined, right? So essentially we have to analyze the limit. Such that we first will break it down, right, because we have a sum we can write two limits. We have limited as x approaches 0 from the right of 2 divided by X rates the power of 1/3, which according to the properties of exponents can also be written as the cube root of X. And then we are adding limit as X approaches 0 from the right. Of 3 divided by Cube root again because our exponent is 1/3 of X + 8. Let's begin analyzing the first limit. So as we already know, if we perform direct substitution, we get 2 divided by 0, which is undefined, right? But we have to recall that in the context of limits, a non-zero number divided by 0 leads to infinity as a limit, right? So we're dividing 2 by 0 from the right, and essentially this gives us positive infinity. Right? Because taking a cube root of a positive number gives us a positive number. So cube root of 0 is 0. We're dividing 2 by a positive number is 0 from the right. This gives us positive infinity since it's a ratio of 2 positive numbers. For the second limit, we can perform a direct substitution, right? We're going to get 3 divided by. Cube wrote Of 0 + 8, which is just 8. So overall the ratio is 3 divided by 23 halves, and now we can see that we're adding those. So what is infinity plus 3 halves? Well, essentially infinity plus 3 halves is just infinity, right? So we can say that. Our answer is positive infinity, so the final answer corresponds to the answer to C. Thank you for watching.

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

lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) as

a. x → 0⁺

Key Concepts

Limits

Indeterminate Forms

Behavior Near Zero

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