Determine the end behavior of the following transcendental fun...

Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.

$f (x) = - 3 e^{- x}$

Accepted Answer

Welcome back, everyone consider the transcendental function F of X equals five E to the negative X. What is the N behavior of this function as X approaches infinity and negative infinity sketch graph of the function showing asymptotes if they exist. Now, in order to, for us to figure out the end behavior of the function and to sketch the graph, it would be a good idea for us to think about what is happening at both ends. OK. So in other words, essentially we would like to find the limit as X approaches infinity of F FX and the limit as X approaches negative infinity of F FX. And you know what we know what F FX is. So let's just put the function here five E to the negative X. Now let's start with it as it approaches infinity. OK. Let me just uh clean this up here. Now what do we know is happening as essentially our function? The value of X gets larger. That's, that's basically what we want. Well, as X approaches infinity, we know that the exponent function E to the negative X approach is zero because the exponent becomes increasingly negative making the exponential term approach zero. So in other words, this, OK, as X approaches infinity is going to be equal to five E to the negative infinity, which is going to be equal to five multiplied by zero, which equals zero. Now this tells us then that since FFX approaches zero, as X goes to infinity, the X axis Y equals zero is a horizontal asym to it. OK. So let's just make a note of that. That's gonna help us for our sketch. Now, what about, what about uh as X approaches negative infinity? Well now as X approaches negative infinity E to the negative X approaches infinity because the exponent becomes increasingly positive, which makes the exponential term grow without bones. In other words, this is going to be equal to five to the five E to the negative negative infinity, which is five E to the power of infinity, which is going to be equal to infinity. OK. In other words, it's getting bigger. So now this means that as X goes to negative infinity, FFX increases without bound, there is no horizontal Asymptote in this direction. Know that we have this information. There's only one more thing we need to plot our graph. OK? We need to identify a key point and we can find the key point at X equals zero. OK? Because at X equals zero, F of zero equals five E to the power of zero E to the power of zero equals one. So that means F of zero will be five. OK. So that tells us that 05 is under graph. Once we have that, then we can sketch our graph because we know that it's going to be a horizontal Asymptote along the X axis. And we know that it's going to be approaching zero as X approaches infinity. So no, if we try to do this sketch here, OK. Let me just uh let me just do a quick sketch. So say that this is our Y axis. OK. This is our X axis. And as we said, we can make this line. Well, you know what, let me just make the space here. We can use this a dashed line to represent the horizontal Asymptote Y equals zero. OK? And no, like we said, we know that 05, OK, 05 is going to be on our graph. Now, we can simply draw our curve. OK. We can draw our curve here and we know that it's going to be in the form of an Asymptote. OK. In other words, as X gets larger, it approaches zero. While as X gets or sorry, as X approaches negative infinity, then our graph will approach infinity. So this would be a sketch of the curve for our graph. Thanks a lot for watching everyone. I hope this video helped.

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.
$f (x) = - 3 e^{- x}$

Key Concepts

End Behavior of Functions

Limits

Asymptotes

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