Take the set of all sequences {x_{n}} such that
lim x_{n} = b ; for each n, x_{n} is different from b ; for each n, x_{n} is in the domain of f.With each sequence {x_{n}} corresponds an 'image sequence' {f(x_{n})}.
lim f(x) = c or lim f(x) = c x->b bIf there isn't such value c, we say that lim f(x) is not defined.
x.x - 5x + 6 (x - 2)(x - 3) lim --------------- = lim --------------- = 1 3 x - 3 3 x - 3 lim sqrt(x) is not defined -2 2.x^{2} + x 2 lim ------------ = --- +infty 3.x^{2} + 4 3
lim c.f(x) = c . lim f(x) (with c = constant) b b lim |f(x)|= |lim f(x)| b b p p lim f(x) = (lim f(x) ) b b lim pth-root(f(x)) = pth-root(lim f(x) ) b b lim (f(x) + g(x)) = lim f(x) + lim g(x) b b b lim (f(x) . g(x)) = lim f(x) . lim g(x) b b b lim (f(x) / g(x)) = lim f(x) / lim g(x) b b b
lim r = r (for each constant number r) b lim x = b b n n lim x = b (n is positive integer) b lim (1/x) = 0 infty n lim (1/x) = 0 (n is positive integer) infty lim nth-root(x) = +infty +infty lim nth-root(x) = -infty -infty lim 1/nth-root(x) = 0 +infty If T(x) and N(x) are polynomials lim T(x) = T(b) b T(x) T(b) lim ---- = ---- b N(x) N(b) lim sqrt(T(x)) = sqrt(T(b)) b
lim f(x) bThis limit is called 'the left limit of f(x) in b' .
lim f(x) < bAnalogous: f: R -> R : x -> f(x) is a real function and there is a strictly positive real number e such that ]b,b+e[ is part of the domain of the function f.
lim f(x) bThis limit is called 'the right limit of f(x) in b' .
lim f(x) > b
1 lim ------ = - infty < b x - b 1 lim ------ = + infty > b x - b
If lim f(x) not = lim f(x) then lim f(x) is not defined < b > b b
If lim f(x) = lim f(x) = c then lim f(x) = c < b > b b
If lim f(x) = c then we have not always lim f(x) = lim f(x) = c b < b > b
lim (2x^{2} - 6x + 7) = infty lim 2x^{2} .(1 - 3/(2x) + 7/(2x.x) ) = infty lim 2x^{2} . lim (1 - 3/(2x) + 7/(2x.x) ) = infty infty lim 2x^{2} infty
If T(x) = a.x^{n} + ... + l is a polynomial then lim T(x) = lim (a.x^{n} ) infty infty
If T(x) = a.x^{n} + ... + l is a polynomial then And N(x) = b.x^{m} + ... + k is a polynomial then T(x) ( a.x^{n} ) lim ----- = lim ------- infty N(x) infty ( b.x^{m} )
-5x - 81 -5x - 81 1 lim -------------- = lim --------- . ------- > -3 (x + 3)(x - 1) > -3 (x - 1) (x + 3) = (16.5) . (+infty) = (+infty) ---------- -5x - 81 -5x - 81 1 lim -------------- = lim --------- . ------- < -3 (x + 3)(x - 1) < -3 (x - 1) (x + 3) = (16.5) . (-infty) = (-infty)
2x.x - 4x 2x.(x - 2) lim -------------- = lim --------------- > 2 x.x - 4x + 4 > 2 (x - 2)(x - 2) 2x = lim --------- = +infty > 2 (x - 2)Example 2:
2x.x - 4x 2x.(x - 2) lim -------------- = lim --------------- 2 x.x - 5x + 6 2 (x - 2)(x - 3) 2x = lim --------- = -4 2 (x - 3)
sqrt(x-3) -1 lim ------------- = 4 x - 4 (sqrt(x-3) -1)(sqrt(x-3) +1) lim ---------------------------- = 4 (x - 4) (sqrt(x-3) +1) (x - 3 - 1) lim ---------------------------- = 4 (x - 4) (sqrt(x-3) +1) 1 lim ---------------- = 0.5 4 (sqrt(x-3) +1)Example 2:
______________ | 2 1 - \| x - 3 x + 3 lim --------------------- 1 _________ | 2 \| 4 x - 3 - 1 We multiply the numerator and the denominator with the factor F = _____________ _________ | 2 | 2 ( 1 + \| x - 3 x + 3)( \| 4 x - 3 + 1 ) _________ 2 | 2 (1 - x + 3 x - 3) ( \| 4 x - 3 + 1 ) = lim -------------------------------------------------------- 1 _____________ 2 | 2 ( 4 x - 4 ) ( 1 + \| x - 3 x + 3) (1 - x^{2} + 3 x - 3) = lim ---------------------- = ... = 1/8 1 ( 4 x^{2} - 4 )Example 3
______________ _________ 3| 3 3| 2 \| x - 2 x - 3 - \| 2 x - 7 lim ---------------------------------- 2 2 x^{3} + x - 18 We multiply the numerator and the denominator with the factor F = _________________ ___________________________ _____________ 3| 3 2 3| 3 2 3| 2 2 \| (x - 2 x - 3) + \| (x - 2 x - 3) (2 x - 7) + \| (2 x - 7) Then we have for the limit x^{3} - 2 x - 3 - 2 x^{2} + 7 lim ---------------------------- 2 (2 x^{3} + x - 18) . F (x^{2} - 2) (x - 2) =lim ---------------------------- 2 (2 x^{2} + 4 x + 9) (x - 2) . F (x^{2} - 2) 2 =lim ----------------------- = -------- 2 (2 x^{2} + 4 x + 9) . F 25 .3
1 + sqrt(-x) lim -------------- = > -2 x + 2 (1 + sqrt(-x)) 1 lim ---------------.-------- = +infty >-2 1 (x + 2)Example 2:
x^{2} - 5x + 4 lim ------------------- = >3 sqrt(x^{2} - 5x + 6) 1 lim (x^{2} - 5x + 4).------------------- = >3 sqrt( (x-2)(x-3) ) (-2).(+infty) = -inftyExample 3:
x^{2} - 5 x + 4 lim ----------------- = <3 _____________ | 2 \| x - 5 x + 6 1 lim (x^{2} - 5x + 4).------------------- = <3 sqrt( (x-2)(x-3) ) is not defined
________ | 2 \| x + 1 + 3 x lim ------------------- = +infty 2x - 5 _________ | -2 (\| 1 + x + 3) x lim ----------------------- = +infty x.( 2 - 5/x) _________ | -2 (\| 1 + x + 3) 4 lim -------------------------- = --- = 2 +infty ( 2 - 5/x) 2Example 2:
________ | 2 \| x + 1 + 3 x lim ------------------- = -infty 2x - 5 _________ | -2 x(3 - \| 1 + x ) lim ------------------------ = -infty x.( 2 - 5/x) _________ | -2 (3 - \| 1 + x ) 2 lim -------------------------- = --- = 1 -infty ( 2 - 5/x) 2
________________ | 2 lim ( \| 4 x + 3 x - 1 + 2 x ) = -infty ________________ ________________ | 2 | 2 (\| 4 x + 3 x - 1 + 2 x)(\| 4 x + 3 x - 1 - 2 x) lim ----------------------------------------------------- = -infty ________________ | 2 (\| 4 x + 3 x - 1 - 2 x) (4x^{2} + 3x - 1) - 4x^{2} lim ------------------------------------- = -infty ________________ | 2 (\| 4 x + 3 x - 1 - 2 x) ( 3x - 1) lim -------------------------------- = -infty ________________ | 2 (\| 4 x + 3 x - 1 - 2 x) x ( 3 - 1/x) lim -------------------------------- = -infty _____________ | 3 -2 (- | 4 + - - x - 2) x \| x ( 3 - 1/x) 3 lim -------------------------------- = ---- -infty _____________ 4 | 3 -2 (- | 4 + - - x - 2) \| xExample 2:
_____________ | 2 lim ( 5 + \| 4 x - x + 3 + 2 x ) -infty _____________ | 2 = lim (2 x + \| 4 x - x + 3 ) + 5 -infty _____________ _____________ | 2 | 2 (2 x + \| 4 x - x + 3 )(2 x - \| 4 x - x + 3 ) = lim --------------------------------------------------- + 5 -infty _____________ | 2 (2 x - \| 4 x - x + 3 ) x - 3 = lim ------------------------------ + 5 -infty _____________ | 2 (2 x - \| 4 x - x + 3 ) x( 1 - 3/x ) = lim ------------------------------ + 5 -infty _________________ | 2 x (2 + \| 4 - 1/x + 3/x ) ( 1 - 3/x ) = lim ------------------------------ + 5 = 1/4 + 5 -infty _________________ | 2 (2 + \| 4 - 1/x + 3/x )
f(x) is continuous in b <=> lim f(x) = f(b) b
f(x) is left continuous in b <=> lim f(x) = f(b) < b
f(x) is right continuous in b <=> lim f(x) = f(b) > b
f is continuous in b => lim f(x) = f(b) b g is continuous in b => lim g(x) = g(b) b So, lim (f(x) + g(x)) = lim f(x) + lim g(x) = f(b) + g(b) b b bQ.E.D. ..
f(x) is continuous in an interval [a,b] <=> f(x) is continuous in each element of [a,b]
(b-a) ------ 2^{n}
(b-a) lim (b_{n}-a_{n}) = lim ----- = 0 => lim a_{n} = lim b_{n} =c_{1} = c_{2} = c infty infty 2^{n}
Now, Since f(x) is continuous in c , lim f(x) = f(c) cHence, for each sequence {x_{n}} with limit c, the sequence {f(x_{n})} has limit f(c). Thus,
lim f(a_{n}) = f(c) = lim f(b_{n}) infty inftyAll terms f(a_{n}) are < 0 , thus f(c) =< 0 .