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Question Number 161516 by naka3546 last updated on 19/Dec/21
f′(a)  is  derivative  of  function  f(a) .  lim_(h→0)   ((f(a−2h^2 )−f(a+h^3 ))/h^2 )  =  ?
$${f}'\left({a}\right)\:\:{is}\:\:{derivative}\:\:{of}\:\:{function}\:\:{f}\left({a}\right)\:. \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{{f}\left({a}−\mathrm{2}{h}^{\mathrm{2}} \right)−{f}\left({a}+{h}^{\mathrm{3}} \right)}{{h}^{\mathrm{2}} }\:\:=\:\:? \\ $$
Commented by cortano last updated on 19/Dec/21
 =lim_(h→0)  ((−4hf ′(a−2h^2 )−3h^2 f ′(a+h^3 ))/(2h))   = lim_(h→0)  ((−4f ′(a−2h^2 )−3hf ′(a+h^3 ))/2)   = −2f ′(a)
$$\:=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{4}{hf}\:'\left({a}−\mathrm{2}{h}^{\mathrm{2}} \right)−\mathrm{3}{h}^{\mathrm{2}} {f}\:'\left({a}+{h}^{\mathrm{3}} \right)}{\mathrm{2}{h}} \\ $$$$\:=\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{4}{f}\:'\left({a}−\mathrm{2}{h}^{\mathrm{2}} \right)−\mathrm{3}{hf}\:'\left({a}+{h}^{\mathrm{3}} \right)}{\mathrm{2}} \\ $$$$\:=\:−\mathrm{2}{f}\:'\left({a}\right) \\ $$
Commented by naka3546 last updated on 19/Dec/21
How  to  get  the  result  without  L′Hospital,  sir?  Thank  you.
$${How}\:\:{to}\:\:{get}\:\:{the}\:\:{result}\:\:{without}\:\:{L}'{Hospital},\:\:{sir}? \\ $$$${Thank}\:\:{you}. \\ $$

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