Lim [(X^2)-7X]/X^2

A student wrote us with this question:

Hello. what about this [(X^2)-7X]/X^2 ?

Thanks for asking! We’re going to guess that the question is asking for \displaystyle{\lim_{x \to 0}} since the problem statement didn’t specify. But, if that’s not correct, please write in again and we’ll address the actual problem.

Going with the limit as x \to 0, we of course first try Substitution:

\lim_{x \to 0}\frac{x^2 -7x}{x^2} = \frac{0 - 0}{0} = "\frac{0}{0}"

which is “indeterminate,” meaning it could be anything. We need to do more work to see.

So let’s try our tactic of “Use Algebra to Find a Limit”:

\begin{align*} \lim_{x \to 0}\frac{x^2 -7x}{x^2} &=\lim_{x \to 0}\left( \frac{x^2}{x^2} - \frac{7x}{x^2}\right) \\[8px] &= \lim_{x \to 0}\left( 1 - \frac{7}{x}\right) \\[8px] &= 1 - \lim_{x \to 0}\frac{7}{x} \\[8px] &= 1 - "\frac{7}{0}" \end{align*}

Since this result contains \frac{\text{a non-zero number}}{0}, this limit does not exist, which is the answer.

\lim_{x \to 0}\frac{x^2 -7x}{x^2} \quad \text{does not exist (DNE)}

We discuss the situation for almost this same equation on our page about using Substitution as a tactic.

The function itself, rewritten as f(x) = 1 - \frac{7}{x}, might make you think of a hyperbola (the -\frac{7}{x} part), simply shifted up 1 unit by that initial “1.” And if you think about what the hyperbola \frac{7}{x} does as x \to 0, the picture in your head might suggest that the limit at x=0 doesn’t exist. A quick check on Desmos (one of our favorite tools! and free for you to use) shows that this is the case:

We discuss limits like this that do not exist on our page " Some Limits That Do Exist; Some That Do Not" (note example function #6 in particular), and then we discuss vertical asymptotes (which your function has at x = 0) on our Vertical Asymptotes page.

We hope that helps. And again, if we didn’t guess correctly about what your question actually was, please write again and let us know! :slightly_smiling_face: