Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .

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Hi everyone, let's take a look at this practice problem. This problem says determine whether the given statement is true or false. And given that the limit as X purchases infinity of the quantity of 1 + 5 divided by X in quantity rates to the X is equal toe raised to the quantity of 1 divided by 5 in quantity. And we're given two possible choices as our answers for choice A we have true, and for choice B, we have false. Now, we need to determine whether the given statement is true or false. So, we're gonna start off by taking the natural log of both sides of our equation. In doing so, we'll have the limit as X approaches infinity. Of the natural log of the quantity of 1 + 5 divided by X in quantity raised to the power X is equal to 1 divided by 5. So, what we need to do is evaluate the limit on the left-hand side and see if it is equal to 1 divided by 5. So we'll have the limit. As x approaches infinity of the natural log of the quantity of 1 + 5 divided by X in quantity rates to the power X, and we're going to rewrite this expression using the properties of logarithms. So we're going to be using our power rule for logarithms. So we'll have the limit as X approaches infinity. Of X multiplied by the natural log of the quantity of 1 + 5 divided by X in quantity. And what we're going to do now is actually generate a fraction by rewriting this to be equal to the limit, as X approaches infinity of the natural log of the quantity of 1 + 5 divided by X in quantity, divided by the quantity of 1 divided by X in quantity. And the reason why we wanted to generate a fraction is first, when we try to evaluate this limit by direct substitution, this is going to be equal to the natural log of the quantity of 1 plus 5 divided by infinity and quantity, divided by the quantity of 1 divided by infinity and quantity, which is equal to 0 divided by 0, which is an indeterminate form. But, but since we have a fraction, we can actually use Lobital's rule to evaluate this limit. So that was the advantage of converting our expression into one with a fraction. So, we call for Lopital's rule, that we need to take the derivative with respect to X of both our numerator and denominator, and then re-evaluate the limit. So, this is going to be equal to the limit. As x approaches infinity of the derivative with respect to X of the natural log of the quantity of 1 + 5 multiplied by X rates to the -1 power, so here we've just rewritten 1 divided by X as X rates to the -1 power. And then this is divided by. The derivative with respect to X. Of the quantity of X-rays to the -1 power. So taking the derivatives, this is going to be equal to. The limit as X per infinity. And in the numerator we'll have the derivative with respect to X of the natural log of the quantity of 1 + 5 multiplied by X-rays to the minus 1 power in quantity. I recall that when you take the derivative of the natural log function, you get 1 divided by the argument, so we'll have 1 divided by the quantity of 1 + 5 multiplied by X-rays to the -1 power in quantity. And then that gets multiplied by the derivative of the argument, since we have to apply our chain rule. So we'll have the derivative with respect to X of the quantity of 1 + 5 multiplied by X-rays to the -1 power. And so when I take that derivative, I will have -5 multiplied by X-rays to the -2 power. Then that gets divided by the derivative with respect to X of X-rays to the -1. And when I take that derivative, I get minus X-rays to the -2. Now, to simplify our expression here, we can see that the factor of X raised to the -2 power cancels out, and then simplifying the rest. This is going to be equal to the limit, as X approaches infinity. Of 5, divided by the quantity of 1 plus 5 divided by X. So here we've just rewritten X raises to the minus 1 power as 1 divided by X again. So, evaluating this limit by direct substitution, this is going to be equal to 5, divided by the quantity of 1 plus 5 divided by infinity. Which is equal to 5. Divided by the quantity of 1 plus 0. Which is equal to 5, but that is not equal to 1 divided by 5, which is what we were looking for. So, since these two values are not equal, that means that our statement is false. So that is the answer that we are looking for. When we come back up to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .

Key Concepts

Limits

Exponential Functions

The Definition of e

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