Lec 6

Limits Def We Wute Iim fex)=LCH) if we can make values of the output of fecc arbotrarly close to by taking x a x sufficently close to a but /a Re f might not be defined in a ex him x) D problem -> /im inx x x - 0 x - 1 . Answer Some In are difined only on one side 20> = UNis of a I cannot guess , use wres Def Im fe approches from left) x - a If we can make values of fec arbitratrally close (max) 3 to L , by taking i suffcenty close to a at the im ( 3 immest e xLa x - 25 im If we can make values of fec arbitratrally close x - 2 to L , by taking a suffcenty close to a but Rules - ( # * 1 lim C = C x - a exercise 2 11m x = a Find (im 3x3- x x 2 x - a x - I 3 . Im(fex = g(x) limfx) + limg(x) -lim 3-lims2+ limx + lim-2 x a x a x a x - > - 1x > 1 x - 1x - - 1 ↓ ⊥ m (Axgr) =(imPex) limgx 3) ! ) = ) =) x - a x - > a x - a = 7 · im Rex x + a 5 im test = - 7 + 1 1 - 2 m g(x) = - 9 x -> a exercise now factor 3 Im (roc)"=(m) " : im i e x - a · im ~res x -> - - i . m"VE=(at)" line x-1 - 1 1(1 + x) ( - 1 1) 2 x - 1 - him I "m -x2 x - a -m him-1 I 11 x -> 1- > 1 x - - x + - 1 =1 - 1 = Eraseries , nate I -

exercise Def we wate him fix) s if we can and me I make the values arbitrally large positive , by taking x sumcently close to a , but ecfa - m - factor (x 4) x - 4 + 2 + 2 x 4 x((-5 + 2) imfe e (5 2) ! "4 x 2 = - is fo filled lim =( 1)(X 2) ᶑ x - > 2 Def we say =a Is a v A if one at least lim g(x) DNE 11m 05 m fix) os lim f(x) 00 x - 2- x -> a x - a lim x - 2 + = 0 m fix) os lim f(x) loo - x - > at x - a you have to say Def we write Im Rs= L If we can x -> - - 2 make the values of the output vales of fex arbitratrally close to L , by taking x sufficently large positive In the same him fix) = L - ...... - im f(x) 2 m fex) = - 3 x - > - 00 - x - 700 ↑ - - - Def we say I is continuos at a if m (x) = f( ) we say its continuos If It Is example linear In , poly , Sin , cos , exp , log continous at every put in D Any combo of continus will be If you can draw grapm wo lifting controus (+ , , , :) pencil FACT · m F(g(x)) = f (lim gc c) if is continous x = a x - a