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

Question

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

63. lim (x → ∞) ((x + 2)/(x - 1))^x

Accepted Answer

Hello. In this video we want to find the given limit. Now the limit that is given to us is the limit as x approaches infinity of the quantity 2 x plus 3 divided by 2 x minus 1, all raised to the power of x. Now, when we are approaching this type of limit, notice that we have a variable in the exponential value. The first thing we want to do is you want to try to remove the variable from the exponential value, and in order to do that, we will need to use our natural logarithm properties. To start, let's say that the output of this limit is going to be some generic limit L. Then what we can go ahead and do is we can take the natural logarithm of both sides of this equation. That will allow us to rewrite the equation limit, and I will be reordering in this case, but we are going to have the natural logarithm of the generic limit L equal to the limit as x approaches infinity of the natural logarithm of 2 x plus 3 divided by 2 x minus 1, all raise to the power of x. Now, if we were to use the exponential property for the natural logarithm, we can bring the X to the front of the natural logarithm as a coefficient. That would leave us with the natural logarithm of L equal to the limit as x approaches infinity of X multiplied by the natural logarithm of 2 x plus 3 divided by 2 x minus 1. Now, let's go ahead and just focus on the right hand side for now. Now if we were to apply the given limit, we would end up with infinity multiplied by 0, but this is what we call an indeterminate form. So in order to simplify the limit, we will need to apply Lahopito's rule, but Lahopito's rule only works on rational functions. Focusing on the right hand side for now, we can express the right hand side of this function as a rational function if we were to rewrite it as a complex fraction. On the right hand side we can rewrite this as the limit as x approaches infinity of the natural logarithm of 2 x plus 3 divided by 2 x minus 1 all divided by 1 divided by x. We can further simplify this by using the quotient property for the natural logarithm. That will leave us with the limit as x approaches infinity of the natural logarithm of 2 x plus 3 minus the natural logarithm of 2 x minus 1, all divided by 1 divided by x. Nowhopito's law states that we want to take the derivative of the numerator and denominator independently. Now, if we were to take the denomin derivative of the numerator, we will have to apply the derivative identity of each of the natural logarithms independently. So that will leave us with the limit as x approaches infinity of 2 divided by 2 x plus 3 minus 2 divided by 2 x minus 1, and this will be divided by the derivative of 1 over x, which is negative 1 divided by x 2. Now if we were to combine the quantities in the numerator together, that will leave us with the limit as x approaches infinity of -8 divided. By 4 x 2 plus 4 x minus 3, and this will all be divided by -1 divided by x 2. And further simplifying this is going to leave us with the limit as x approaches infinity of 8 x 2 divided by 4 x 2 plus 4 x minus 3. So what we are doing now is we are taking an infinite limit of a rational function, but notice that the leading powers of the numerator and denominator are exactly the same. The trick here is that when you apply the the. When you apply The properties of the infinite limit, this was simplified to just be the coefficients ratio. So, the limit of this value is going to simplify to be 8 divided by 4. And that is going to leave us with 2. So this is going to be the simplified right-hand side of our original function by using the Hopito's law, but keep in mind that this was originally equal to a generic limit we stated earlier. So plugging this back into the original limit, leaves us with the natural logarithm of L equal to 2. In order to remove the limit from the natural logarithm, we will exponentiate both sides of this equation. And exponentiating both sides of the equation leaves us with the limit l equal to E2. This is going to be the value of the limit for the given problem. And with that being said, the overall solution to this problem is going to be option B. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Indeterminate Powers and ProductsFind the limits in Exercises 53–68.63. lim (x → ∞) ((x + 2)/(x - 1))^x

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Indeterminate Powers and Products
Find the limits in Exercises 53–68.
63. lim (x → ∞) ((x + 2)/(x - 1))^x