Determine the following limits at infinity.

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Question

Determine the following limits at infinity.

lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the limits given below. The limit as X approaches infinity of 9 to the power of X, the limit as X approaches negative infinity of 9 to the power of X. And the limit as X approaches infinity of 9 to the power of negative X. Awesome. So it appears for this particular prong we're asked to solve for 3 different limits, or rather we're asked to evaluate three different limits. So that means we're solving for 3 separate answers. Awesome. So now that we know that we're ultimately trying to solve for the evaluation or we're trying to evaluate 3 separate limits, meaning we're gonna have 3 separate answers. Let's read off our multiple choice answers to see what our final answer set might be, and note that they're in order as they're listed in the problem itself, so it's the limit as X approaches infinity of 9 to the power of X is equal to some value. And then the next one is the limit as X approaches negative infinity of 9 to the power of X, and lastly limit as X approaches infinity of 9 to the power of negative X. They're all equal to some value. So A is positive infinity, positive infinity, positive infinity, B is positive infinity, positive infinity, and 0. C is 00, and 0. And D is positive infinity 0 and 0. Awesome. So first off, we need to solve these limits and we will approach each one step by step and to keep things simple, let us call our like call the first limit number one. Our second limit, #2, and our third limit, number 3. And let's start to solve for limit 1 1st, so we're going to be focusing on the limit as X approaches infinity of 9 to the power of X. So first off, for this particular limit, we need to recall and recognize that as X approaches infinity, 9 to the power of X will grow without bound because 9 is greater than 1. So like this is important to note since 9 is greater than 1, so therefore we can go ahead and as a result, we can go ahead and write. That the limit as X approaches infinity of 9 to the power of X will be equal to infinity, and that is our first answer. Hooray, we did it. And then once again, this is for our first limit. Awesome. So now moving on to our second limit. So, right now, we need to evaluate our second limit, which our limit for this particular condition is gonna be the limit. As I mean this is where x approaches negative infinity for this case of 9 to the power of X. Awesome. So first off, for this particular limit, we need to recall and note that as X approaches negative infinity, 9 of X will approach 0. So let's make a note of this. So 9 of X will approach 0 as x approaches negative infinity. Awesome. So, and this is the case because any number greater than 1 raised to a negative infinity power will approach 0. So therefore, without a doubt, we can go ahead and write that the limit as X approaches negative infinity of 9 to the power of X will be equal to 0 as a result. Fantastic. We are on a roll here. So that is our final answer for our 2nd limit. Hooray, we did it. So now we can focus on our third and final limit. So, as for this particular limit, we should recall that we're dealing with the limit as X approaches infinity of 9 to the power of negative X. OK. So for this particular limit, we need to recall and recognize that 9 of negative X is the same as, so let's make a little note here. So we could say that 9 of negative X is also equal to 1 divided by 9 of X. And let us also note and recall that as X approaches infinity, so as X approaches positive infinity, 9 X will grow without bounds, so that will mean that 1 divided by 1 divided by the power of X will approach 0. So therefore, without a doubt, we can go ahead and write that the limit as X approaches infinity of 9 of negative X is going to be equal to 0, and that is our final answer for our third and last limit. Hooray, we did it. So in summary, let's quickly recap here. The limit as x approaches infinity of 9 of X is equal to infinity, and the limit as X approaches negative infinity of 9 of X is equal to 0. And finally, the limit as X approaches infinity of 9 of negative X is equal to 0. And this corresponds with multiple choice answer letter D, which once again states that the limit as X approaches infinity of 9 to the power of X is equal to infinity. The limit of X as X approaches negative infinity of 9 of X is equal to 0. And note that's for negative infinity. And lastly, the limit as x approaches positive infinity of 9 to the power of negative X is equal to 0. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Determine the following limits at infinity.

lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

Key Concepts

Limits at Infinity

Exponential Functions

Behavior of e^t and e^−t

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