l'Hospital's rule

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  1. Evaluating the limit of the form:
    lim\limx →c c f(x)g(x)\frac{f(x)}{g(x)}
  1. Evaluating the limit.
    1. lim\limx →1 1 lnxx1\frac{\ln x}{x-1}
    2. lim\limx → \infty lnxx1\frac{\ln x}{x-1}
Topic Notes
Remember that one tricky Limits section that required intense algebraic manipulation to avoid getting 0/0 or infinity/infinity limits? We will now revisit it again, but with the knowledge of derivatives. In this section, we will learn how derivatives enable us to efficiently evaluate the limits of a function using the "L'Hospital's rule".
Note *l'Hôpital's Rule applies to 2 types of indeterminate forms:

type 00\frac{0}{0} (that is, lim\limx →c c f(x)=0f(x)=0 and lim\limx →c cg(x)=0g(x)=0)
type \frac{\infty}{\infty} (that is, lim\limx →c c f(x)=±f(x)=\pm \infty and lim\limx →c cg(x)=±g(x)=\pm \infty)

Then according to l'Hôpital's Rule: lim\limx →c c f(x)g(x)=\frac{f(x)}{g(x)}= lim\limx →c c f(x)g(x)\frac{f'(x)}{g'(x)}