Evaluate Derivative $\lim_{x \to 1}\frac{10x-1.86x^2 - 8.14}{x - 1}$

I've already evaluated the limit using the $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ definition of a limit, but now I'm curious as to how you would factor the numerator in $\lim_{x \to 1}\frac{10x-1.86x^2 - 8.14}{x - 1}$ to get rid of the $x-1$ in the denominator.

A solution to the factoring problem I encountered was using Polynomial Long Division, but I'm curious for more ways to factor it.

  • $\begingroup$ is $t$ in the numerator supposed to be an $x$? $\endgroup$ – mm-aops Jul 31 '14 at 14:06
  • $\begingroup$ Yes, thanks for pointing that out. $\endgroup$ – Kermit the Hermit Jul 31 '14 at 14:07
  • $\begingroup$ Use the formula for the roots en.wikipedia.org/wiki/Quadratic_equation. This gives the factorisation. $\endgroup$ – gammatester Jul 31 '14 at 14:12

The solutions of $-1.86x^2+10x-8.14=0$ are $x=1$ and $x=4.37634$.

So,the limit is equal to this one:

$$\lim_{x \to 1} \frac{-1.86(x-1)(x-4.37634)}{x-1}=\lim_{x \to 1} (-1.86(x-4.37634))$$

  • 2
    $\begingroup$ That's cool, I never really looked at it like that, yet it seems so obvious. Much appreciated, evinda. $\endgroup$ – Kermit the Hermit Jul 31 '14 at 14:43

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