Indeterminate Powers and Products
Find the limits in Exe...

Question

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

53. lim (x → 1⁺) x^(1/(1 - x))

Accepted Answer

Welcome back, everyone. Evaluate the limit as X approaches 0 of 1 + 3 X. Ras the power of 1 divided by X. It says E cubed, B E C E to the power of 13, and D 3. Let's begin this problem by rewriting the limit. Limit as X approaches 0 of 1 + 3 X to the power of 1 divided by X, and let's attempt the direct substitution. In Cs, we're going to get 1 + 3 multiplied by 0, which is 1, and our exponent, 1 divided by 0 is going to approach infinity. Once the power of infinity is an indeterminate form. To work with this in determinant form, one of the ways is to first of all, assign L to the limit. We want to find out. And what we're going to do is take the natural log of L to apply the properties of logarithms and extract. The exponent using the power rule. So LN of L is going to be equal to the limit as X approaches 0 of LN 1 + 3 X raises the power of 1 divided by X, which now allows us to bring down the exponent. And get a fraction or basically a rational function. So we get the limit as X approaches 0 of. LM 1 + 3 X multiplied by 1 divided by X. And now, if we attempt the direct substitution, we got a line of 1 + 0, which is a line of 1, and that's 0. Divided by X, which is 0. Now, 0 divided by 0 is an indeterminate form as well, but for this form, we can apply Lapito's rule. According to Lapito's rule, we want to take the derivative of. Of the numerator and divided by the derivative of the denominator. Let's valuate each derivative. In the numerator, we have a composite function. We first will differentiate natural log with respect to the inner function to get 1 divided by 1 + 3 X and we multiply by the derivative of the inner function, right, according to the chain rule. And then we, we divide by X derivative, which is 1. So, division by 1 doesn't affect anything. We can just remove one and focus on the remaining terms, right? The derivative of 1 is 0, the derivative of 3 X is 3 multiplied by 1, which is 3. So we get limit as X approaches 0 of 3. Divided by 1 + 3 X. We can now attempt the direct substitution again to get 3 divided by 1 + 3 multiplied by 0. And this is equal to 3. So we have got a line of L, meaning the limit L itself is going to be E, raised to the power of 3 or basically E cubed. So the correct answer for this problem corresponds to the answer choice A. Thank you for watching.

Indeterminate Powers and ProductsFind the limits in Exercises 53–68.53. lim (x → 1⁺) x^(1/(1 - x))

Watch next

Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))