Infinite Limits

Find the lim...

Question

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 4 / x²/⁵

Accepted Answer

Welcome back, everyone. Find the limit as x approaches 0 for the function F of X equals -12 divided by X raises to the power of 3/5. Were given 4 answer choices A says infinity, B 0, C negative infinity, and D does not exist. So first of all, let's analyze the limit. Limit as x approaches 0 of negative 12 divided by X to the power of 3/5. We can essentially rewrite it as follows. This is limit as x approaches 0 of -12 divided by. Now, what is X rates the power of 3/5? Using the properties of exponents, we can say that this is the 5th root of X cubed. Now if we just substitute 0 in, we're going to get division by 0 and a number divided by 0. Well this generally leads to a limit of infinity, but we want to make sure that our left sided limit is equal to the right sided limit, right? So what we're going to do is basically analyze what happens as X approaches 0 from the left first. We're going to get -12 divided by 5th root of XQ. Now we cannot directly substitute, but we can essentially show an attempted substitution. So we're going to get -12 divided by the fifth root of 0 from the left cubed, right? So this is going to give us The 5th root of 0 from the left, right? If we cube a negative number, we're going to get a negative number, and if we take an odd root of a negative number, it is going to be a negative number. So this allows us to visualize in brackets what's happening here. We are essentially dividing 12, a negative number by 0 from the left, also a negative number. So this ratio will give us positive infinity because this is a ratio of 2 negative numbers. Similarly, let's analyze the limit as X approaches 0 from the right of -12, divided by the fifth root of X cubed. In this case, we can perform the same analysis which gives us -12 divided by the fifth root. Of 0 from the right cube, which gives us 0 from the right. In other words, if we are cubing a positive number, it gives us a positive number, and taking the fifth root of 0 from the right gives us 0 from the right as well, still a positive number. So in this case, we have -12 divided by 0 from the right, a ratio between a negative and a positive number, which gives us a negative number, negative infinity, right? Now these two limits are not equal to each other, meaning the limit as X approaches 0 itself does not exist, right? Because according to the limit existence theorem, the two-sided limits must be equal to each other. To guarantee that the limit as X approaches the point itself exists, right? So we can conclude that the correct answer to this problem corresponds to the answer choice D does not exist. Thank you for watching.

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 4 / x²/⁵

Key Concepts

Infinite Limits

Behavior of Rational Functions

Power Functions and Exponents

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