This is scary: "if you plug in zero you get a 0/0 situation at x/abs(x)." Why would you "plug in" ANYTHING without first determining if it exists? Never do that. You do not get a "0/0 situation", because you never would think of making such a substitution with a value your not sure is in the

Domain. Clearly you are not sure. Don't do it!

You just have to wade through it. There is nothing magic. You don't have to expand first, but I decided to do it.

Expand [tex](x+|x|)^{2} + 1 = x^{2} + 2\cdot x\cdot |x| + |x|^{2} + 1 = 2\cdot x^2 + 2\cdot x \cdot |x| + 1[/tex]

For x > 0, |x| = x and [tex]f(x) = 4\cdot x^{2} + 1[/tex] and [tex]f`(x) = 8\cdot x[/tex]

For x < 0, |x| = -x and [tex]f(x) = 1[/tex] and [tex]f`(x) = 0[/tex]

You decide what to do for x = 0.

1) Does the limit exist from both sides?

2) If so, is it the same from both sides?

There's a reason why we make and teach definitions. Use them!

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