How to use the alternate definition to find the derivative of f(x)=sqrt(x+3) at x=1?

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Here are a stranger ways you can do the season balance for the derivative. Both methods complicate "rationalizing the numerator" (not the denominator) as a finesse to aid you consider the seasons.

##f'(1)=lim_{h->0}frac{f(1+h)-f(1)}{h}##

##=lim_{h->0}frac{sqrt{4+h}-2}{h}cdot frac{sqrt{4+h}+2}{sqrt{4+h}+2}##

##=lim_{h->0}frac{4+h-4}{h(sqrt{4+h}+2)}=lim_{h->0}frac{1}{sqrt{4+h}+2}=frac{1}{4}##

OR

##f'(1)=lim_{x->1}frac{f(x)-f(1)}{x-1}=lim_{x->1}frac{sqrt{x+3}-2}{x-1}##

##=lim_{x->1}frac{x+3-4}{(x-1)(sqrt{x+3}+2)}##

##=lim_{x->1}frac{1}{sqrt{x+3}+2}=frac{1}{4}##

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