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# Find the limit or show that it does not exist. $\displaystyle \lim_{x \to \infty} \frac{x^4 - 3x^2 + x}{x^3 - x + 2}$

## $\infty$

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all right, we want to find a limit of this expression as X approaches positive infinity. Since X is going to be approaching positive infinity, it's going to be positive and it's going to be very large and then larger and even larger. So when we look at the numerator, X to the fourth minus three X squared plus x. Even though we have three terms uh as X gets very very large uh one of these terms, the one with the highest power is going to be the dominant term, so the numerator is going to be dominated by extra fourth uh because as X is very large, exit 1/4 will be extremely large three times X to the second won't be anywhere near uh you know, as large as X to the fourth. So even though we're subtracting three X squared uh this whole numerator really is going to be dominated by the highest power backs D x to the fourth term. Likewise, the denominator is going to be not is going to be dominated by the X to the third term. So this entire expression is basically going to be dominated by X to the 4th over X to the third as X approaches positive infinity Now exit the 4th divided by extra 3rd is really x. Okay, so as X approaches positive infinity. This entire expression is basically going to behave as the value of X itself would. So it's X gets a very very large X or this expression this limit is also going to get very very large. So as X approaches infinity, we expect this function to approach infinity. Here on the graph. I have the function graft. And you can clearly see that as as X approaches positive infinity, the function is approaching positive infinity. Okay, we can zoom out and you can see the more and more you move to the right as X approaches positive infinity are the function is getting higher and higher. So as X approaches positive infinity, our function approaches positive infinity, and that's why our function has an infinite limit. The limit of our function as X approaches infinity is infinity.

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