To be answered: never mind number3

/I'opic: Limit Theorems We are now ready to list down the basic theorems on limits. We will state eight theorems. These will enable us to directly evaluate limits, without need for a table or a graph. In the following statements, c is a constant, and f and g are functions which may or may not have c in their domains. ' l. The limit of a constant is itself. If k is any constant, then, liquc k = k For example, I. liquc 2 = 2 2. limxsc 8 = -8 3. lint,\"C 0.25 = 0.25 For the remaining theorems, we will assume that the limits of f and g both exist as x approaches 6 and that they are L and M, respectively. In other words, lim,-cf(x) = L, and lim,-eg(x) = M 2. The Constant Multiple Theorem: This says that the limit of a multiple of a function is simply that multiple of the limit of the function. lim,-ck.f(x) = klim,-cf(x) = kL Examples: If limx-cf(x) =4, then 1 . lim,-c 8. f(x) = 8limx-cf(x) = 8.4 = 32 6. The anger power got a function is just that 2. limx-c-11f(x) =-11 limx-cf(x) = -11.4 =-44 3. limx-cf(x) = limx-cf(x) =4=6 3. The Addition Theorem: This says that the limit of a sum of functions is the sum of the limits of the individual functions. Subtraction is also included in this law, that is, the limit of a difference of functions is the difference of their limits. lim,-c(f(x) + g(x)) = limx-cf(x) + limx-cg(x) =L+M limx-c(f(x) - g(x)) = limx-cf(x) - limy-cg(x) = L- M 4. The Multiplication Theorem: This is similar to the Addition Theorem, with multiplication replacing addition as the operation involved. Thus, the limit of a product of functions is equal to the product of their limits. limx-c(f(x) . g(x)) = limx-cf(x) . limx-cg(x) = L. M Examples: 1. Let limx-cf(x) =4 and limx-cg(x) = -5, then limx-c(f(x) . g(x)) = limx-cf(x) . limx-cg(x) = (4)(-5) = -20 Remark 1: The Addition and Multiplication Theorems may be applied to sums, differences, and products of more than two functions. Remark 2: The Constant Multiple Theorem is a special case of the Multiplication Theorem. Indeed, in the Multiplication Theorem, if the first function f(x) is replaced by a constant k, the result is the Constant Multiple Theorem. 5. The Division Theorem: This says that the limit of a quotient of functions is equal to the quotient of the limits of the individual functions, provided the denominator limit is not equal to 0. lim /(2 limx-cf(x) x-c g(x) limx-cg(x) provided M # 0 Examples: 1. Let limx-cf(x) =4 and limx-c g(x) =-5, then lim /(x) limx-cf(x) x-c g(x) limx-c g(x) -5 2. If limx-cf(x) = 0 and limx-c g(x) = -5, then lim (2) limx-cf(x) x-c g(x) limx-c 8(x) -= 03. If lim,-e f (x) = 4 and limx-cg(x) =0, then xec 8(x) limper 9(x) lim [(2) x-c g(x) Since M= 0, then lim 2 x-c g(x) , DNE 6. The Power Theorem: This theorem states that the limit of an integer power p of a function is just that power of the limit of the function. limx-c(f(x))" = (limx-cf(x))" = LP mite off er enotromil To mine s to timid and is a2 airfl monoonT noitibby ofT Examples, 1. If lim,fox) Then wal and in bobuteni oats at nonosudud enonemu lanbind lim,-(f(x)) = (limx-cf(x))3 enmil north to conoroflib s = (4)3 14 1 =64 03 mil + ()\\ mil = (()+ (x) U.-.mil 2. If limx-cf (x) =4, then limx-c((f(x))= (limx-cf(x))-2 = ( 4 )"- 1 42 had to boubor o'to mamp with audi benton nousisgo off es norible arim 16 7. The Radical/Root Theorem: This theorem states that if n is a positive integer, the limit of the nth root of (0 02- mil a function is just the nth root of the limit of the function, provided the nth root of the limit is a real number. Thus, it is important to keep in mind that if n is even, the limit of the function must be positive, limx-cVf (x) =" limf(x) () | mil -(1x)g. (x.) Uxmil For example, ( E -. ) ( A ) : 1. If limx-cf (x) =4, then 25011514 limx-c Vf(x) =" x-c limf (x) enondonut out nath som to arouborg bus nononul lent of hi mswoordT nousoilqiluM ads a boobat 2. If limx-cf(x) = 4, then it is not possible to evaluate , lime f(x) because then, limx-c VVf(x) =V-4 , that is v-4 is not a real number. ardi to basitoup side of isups at endon to monoup a to tofail and todayse aulT unowood ! noizlaid ofTlimx-cf(x) = L, and limx-cg(x) =M Histhi . 0 = (x)D , mil bus + = (x) tamil H E 2. The Constant Multiple Theorem: This says that the limit of a multiple of a function is simply that (2)0 3+4 multiple of the limit of the function. lim,-ck.f(x) = klimsef(x) = kL (1)8 3+4 nor ,0 =M gonia Examples: mil If limx-e f (x) = 4, then and . 8 32 1. limx-- 8.f(x) = 8limx-cf(x) = 8.4 = 32 .nonomil ordi to timil ordi 10 Tow 3. limx-of(x) =limx-cf(x) =$4=6 U = "((x)\\seymii ) = "((x)V,-y.mil 3. The Addition Theorem: This says that the limit of a sum of functions is the sum of the limits of the individual functions. Subtraction is also included in this law, that is, the limit of a difference of functions is the difference of their limits. "((x) 1 ,.-mil) =((x)V,-mil i (1 ) = limx-c(f(x) + g(x)) = limx-cf(x) + limx-cg(x) =L+M limx-e(f(x) - g(x)) = limx-cf(x) - limx-cg(x)= LM ,-mil l .s "((x) ,-xmil) =((x)0),-mil 4. The Multiplication Theorem: This is similar to the Addition Theorem, with multiplication replacing addition as the operation involved. Thus, the limit of a product of functions is equal to the product of their limits. limx-c(f(x) . g(x)) = limx-cf(x). . limx-9 : mgo9NT 10of1\\sibel 9dT o fimil sill dogsini ovifiedq's al w ti farhi ediste mofoorif am I Examples: ei limil erfi to wor whin erdi bobivorq noifort ords To simil off To boot in oh taut zi noiton 1. Let limx-of(x) =4 and "lim, g(x) -5, then brim ni qood of inshoqmi zi fi aur limx-c(f(x) . g(x)) = limx-cf(x) (x)limcg(x)(x) \\ ,mil = (4)(-5) siqmexs 104 =-20 Remark 1: The Addition and Multiplication Theorems may be applied to sums, differences, and products of more than two functions. (x) ( mil Remark 2: The Constant Multiple Theorem is a special case of the Multiplication Theorem. Indeed, in the Multiplication Theorem, if the first function f(x) is replaced by a constant k, the result is the Constant Multiple Theorem. nori +- = (x) ,-mil 71 .S andmun issi s ion ai f-v ai ish . P-V =(x) \\Vexmil 5. The Division Theorem: This says that the limit of a quotient of functions is equal to the quotient of the limits of the individual functions, provided the denominator limit is not equal to 0. lim / (2) limx-e f(x) x-c g(x) limx-cg(x) provided M # 0 Examples: 1. Let limx-cf(x) =4 and limx-cg(x) = -5, then lim /2 =_ limx-cf(x) x-c g(x) Umzac 9(x) -5 2. If limx-e f(x) =0 and limx-cg(x) = -5, then lim (2 - limx-c /(x) rac g(x) limx-c g(x) 4= 0 -5A. Written Work/s: 1. Given lim f(x) = 3 and lim g(x) = -1, evaluate the following limits. x-+1 (a) lim 2 . f(x) -+1 2. Given lim f(a) = 2 and lim g(x) = -2, evaluate the following limits. -4-1 r-+-1 (a) lim 3. Given lim f(x) = 0 and lim g(r) = 1, evaluate the following limits. 2-+0