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Question Number 10094 by Tawakalitu ayo mi last updated on 23/Jan/17

Prove that if   lim_(x→a)   f_1 (x) = L_1   and    lim_(x→a)    f_2 (x) = L_2  then lim_(x→a)  [f_1 (x)+f_2 (x)] = L_1 +L_2

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\:\:\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\:\mathrm{f}_{\mathrm{1}} \left(\mathrm{x}\right)\:=\:\mathrm{L}_{\mathrm{1}} \:\:\mathrm{and}\:\: \\ $$$$\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\:\:\mathrm{f}_{\mathrm{2}} \left(\mathrm{x}\right)\:=\:\mathrm{L}_{\mathrm{2}} \:\mathrm{then}\:\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\left[\mathrm{f}_{\mathrm{1}} \left(\mathrm{x}\right)+\mathrm{f}_{\mathrm{2}} \left(\mathrm{x}\right)\right]\:=\:\mathrm{L}_{\mathrm{1}} +\mathrm{L}_{\mathrm{2}} \\ $$

Answered by sandy_suhendra last updated on 23/Jan/17

lim_(x→a) [f_1 (x)+f_2 (x)]  =lim_(x→a) f_1 (x) + lim_(x→a) f_2 (x) ⇒ limit theorem  = L_1  + L_2

$$\mathrm{li}\underset{\mathrm{x}\rightarrow\mathrm{a}} {\mathrm{m}}\left[\mathrm{f}_{\mathrm{1}} \left(\mathrm{x}\right)+\mathrm{f}_{\mathrm{2}} \left(\mathrm{x}\right)\right] \\ $$$$=\mathrm{li}\underset{\mathrm{x}\rightarrow\mathrm{a}} {\mathrm{m}f}_{\mathrm{1}} \left(\mathrm{x}\right)\:+\:\mathrm{li}\underset{\mathrm{x}\rightarrow\mathrm{a}} {\mathrm{m}f}_{\mathrm{2}} \left(\mathrm{x}\right)\:\Rightarrow\:\mathrm{limit}\:\mathrm{theorem} \\ $$$$=\:\mathrm{L}_{\mathrm{1}} \:+\:\mathrm{L}_{\mathrm{2}} \\ $$

Commented by Tawakalitu ayo mi last updated on 23/Jan/17

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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