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Numerade Educator

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Problem 13 Easy Difficulty

Evaluate the limit and justify each step by indicating the appropriate properties of limits.

$ \displaystyle \lim_{x \to \infty} \dfrac{2x^2 - 7}{5x^2 + x -3} $

Answer

$\frac{2}{5}$

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Video Transcript

So here we are given a specific limit and were asked to evaluate the limits as X approaches infinity of two, X squared minus seven over five X squared plus x minus three. So as X approaches large numbers, relatively speaking compared to our constant terms, the X axis, The X values in this would be significantly greater. So you can ignore our constant terms. Also, X squared would also be significantly greater than just X for large numbers so we can ignore X. And as a result we can find that our limit would be the limit as X approaches infinity of two, X squared divided by five X. Work. So we can cancel out this X squared And we can find that we just have the limit as X approaches infinity of a constant term. So this would just be equivalent to 2/5. We can also evaluate this by our other method for evaluating limits. Lockable rule. So this would just be taking the derivative of the numerator and the derivative of the nominator. But we require multiple iterations in this case. So the first iteration before X. This would be 10, X plus one. Then we will need another iteration, this would become four and this would become 10 And once again this would be the limit as X approaches infinity for over 10 which is just equal equivalent to 2/5 and simplest form. And this gifts our final answer