Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math 1016: Elementary Calculus with Trigonometry I Ch. 3 Introduction to the Derivative Sec. 3.3: Limits: Algebraic Viewpoints I. The Limit Laws A. Limit Laws If L, M, c and and k are real numbers and let f and g be any 2 functions st lim f ( x ) = L x→c lim g ( x ) = M , then x→c 1. Sum Rule: lim ⎡⎣ f ( x ) + g ( x ) ⎤⎦ = lim f ( x ) + lim g ( x ) = L + M x→c x→c x→c 2. Difference Rule: 3. Product Rule: lim ⎡⎣ f ( x ) − g ( x ) ⎤⎦ = lim f ( x ) − lim g ( x ) = L − M x→a x→a x→a lim ⎡⎣ f ( x ) g ( x )⎤⎦ = lim f ( x ) ⋅ lim g ( x ) = L M x→c x→c x→c 4. Constant Multiple Rule: 5. Quotient Rule: lim ⎡⎣ k f ( x )⎤⎦ = k lim f ( x ) = k L x→c x→c f ( x) L ⎡ f ( x ) ⎤ lim x→c lim ⎢ = = , M ≠0 ⎥ x→c g x lim g x M ( ) ( ) ⎣ ⎦ x→c r and s are integers with no common factors and s ≠ 0 , then r r r s (r) lim ⎡⎣ f ( x ) ⎤⎦ s = ⎡ lim f ( x ) ⎤ = Ls , provided that Ls is a real number. (If s is even, ⎣ x→c ⎦ x→c L>0.) 6. If B. Other Common Limit Rules 1. Constant Function: 2. Identity Function: lim k = k x→c lim x = c x→c 3. If n is a positive integer, then lim x n = c n 4. If n is a positive integer, then lim 5. If m and b are real numbers, then lim [mx + b] = mc + b . x→c x→c n x = n c , provided c > 0 if n is even. x→c C. Limits of Polynomials P(x) is a polynomial, P(x) = an x n + an −1 x n −1 + ... + a2 x 2 + a1 x + a0 , then lim P(x) = P(c) . If x→c Example: lim(7x 3 − 8) = 7(2)3 − 8 = 7(8) − 8 = 56 − 8 = 48 x→2 D. Limits of Rational Functions 1. c is in the domain of Q(x) : If P ( x ) P (c) = . x→c Q(x) Q(c) P(x) and Q(x) are polynomials and Q(c) ≠ 0 , then lim Example: 2. ⎛ x 2 − 4 ⎞ 1− 4 −3 lim ⎜ = = = −1 x→1 ⎝ x + 2 ⎟ ⎠ 1+ 2 3 c is NOT in the domain of Q(x) : If P(x) and Q(x) are polynomials and Q(c) = 0 , then use factor, graphical approach or table. ⎛ x2 − 4 ⎞ 0 ⎛ (x + 2)(x − 2) ⎞ lim ⎜ → = lim ⎜ lim (x − 2) = −4 ⎟⎠ = x→−2 ⎟ x→−2 ⎝ x + 2 ⎠ 0 x→−2 ⎝ x+2 Example: a. fHxL= 1 x2 y b. k ⎛ 1⎞ lim ⎜ 2 ⎟ → → ∞ ; DNE . x→0 ⎝ x ⎠ 0 500 400 300 200 100 -1.0 -0.5 x 2 − y 2 = (x − y)(x + y) Helpful Factoring: x 3 − y 3 = (x − y)(x 2 + xy + y 2 ) x 3 + y 3 = (x − y)(x 2 − xy + y 2 ) E. Examples Evaluate the following limits: 1. 2. 3. 4. lim 4 = x→3 lim x = x→3 lim (t + 9) = t→2 lim (5w − 6) = 2 w →2 ⎧⎪ x 2 − 5x + 4 ; x≠4 5. lim f ( x ) if f (x ) = ⎨ x−4 x →4 ⎪⎩ x - 5 ; x=4 0.0 0.5 1.0 x II. One-Sided Limits A. Informal Definitions 1. Left Hand Limit / Limit from Below: lim f ( x ) = L and say the left-handed limit of f ( x ) as x a. Defn: We write approaches x→c − c [or the limit of f ( x ) as x approaches c from the left] is equal to L if we can make the values of f ( x ) arbitrarily close to L by taking x to be sufficiently close to b. This means that c and x less than c. x approaches c from smaller values. 2. Right Hand Limit / Limit from Above: a. Defn: We write approaches to lim f ( x ) = L and say the right-handed limit f ( x ) as x x→c + c [or the limit of f ( x ) as x approaches c from the right] is equal L if we can make the values of f ( x ) arbitrarily close to L by taking x to be sufficiently close to b. This means that B. Theorem 6 A function c and x greater than c. x approaches c from larger values. f ( x ) has a limit as x approaches c iff it has left-hand and right-hand limits there and these one-sided limits are equal: lim f ( x ) = L ⇔ lim− f ( x ) = lim+ f ( x ) = L , where c, L are real numbers. x→c x→c C. Examples 1. Given x→c ⎧3t + 9 ; t < −2 g(t) = ⎨ 2 ⎩t − 1 ; t > −2 a. lim g(t ) = b. lim g(t ) = t→ 4 t →- 2 lim x −1 = x −1 lim x = lim x= 5. lim x= 6. ⎛ 1⎞ lim+ ⎜ ⎟ = x→0 ⎝ x ⎠ 7. ⎛ 1⎞ lim− ⎜ ⎟ = x→0 ⎝ x ⎠ 2. 3. 4. III. x→1 x→0 + x→0 − x→0 Limits as x → ±∞ In this section we have to modify the definition of a limit so that we can work with infinity. (Recall that ∞ + ∞ = ∞ but ∞ − ∞ is not a defined operation.) A. Informal Definitions 1. Defn: Let f be a fn defined on some interval (a, ∞ ) . Then lim f ( x ) = L means that x →∞ the values of 2. Defn: Let f(x) can be made arbitrarily close to L by taking x sufficiently large. f be a fn defined on some interval (−∞, a) . Then lim f ( x ) = L means that the values of large negative. x →−∞ f(x) can be made arbitrarily close to L by taking x sufficiently B. Theorem If r > 0 is a rational number, then lim x →∞ defined ∀ x , then lim x →−∞ D. Examples Evaluate the following. 1. 2. lim x→∞ lim x→−∞ 1 = x 1 = x −4 3. lim 4. ⎛ 1⎞ lim ⎜⎜13 + 3 ⎟⎟ = y→ −∞ ⎝ y ⎠ 5. ⎛ 4 ⎞ ⎟⎟ = lim ⎜⎜ w→ − ∞ ⎝ 8 − 2 ⎠ w x→∞ ( x − 2 )2 = 1 = 0. xr 1 = 0 . If r > 0 is a rational number s.t. x r is xr E. Rational Functions 1. Procedure: When taking the limit at infinity of a rational function in which you get the form ±∞ , you need to divide all terms by the highest degreed variable in the ±∞ denominator. 2. Examples a. ⎛ 3x + 7 ⎞ ∞ lim ⎜ 2 ⎟ → ⇒⇒ ÷ every term by x 2 x →∞⎝ 4x − 2 ⎠ ∞ Therefore: ⎛ ⎜ ⎛ 3x + 7 ⎞ lim ⎜ 2 ⎟ = lim ⎜ x→∞ ⎝ 4x − 2 ⎠ x→∞ ⎜ ⎝ 3x 7 ⎞ ⎛ 3 7 ⎞ lim ⎛ 3 ⎞ + lim ⎛ 7 ⎞ + 2 ⎟ + ⎜ ⎟ ⎜ ⎟ 2 ⎜ x x 2 ⎟ x→∞ ⎝ x ⎠ x→∞ ⎝ x 2 ⎠ 0 + 0 0 x x = lim = = = =0 2 ⎟ 4x 2 2 ⎟⎟ x→∞ ⎜ ⎛ 2⎞ 4−0 4 4 − ⎜⎝ ⎟ lim ( 4 ) − lim ⎜ 2 ⎟ − 2 x→∞ ⎝ x ⎠ x 2 ⎠ x→∞ x2 x ⎠ b. Evaluate the following. 1.) ⎛ 3x + 1 ⎞ lim ⎜ ⎝ −7x + 2 ⎟⎠ x→ ∞ ⎛ -7x 3 - 8x + 9 ⎞ 2.) lim ⎜ x→ -∞ ⎝ 2x 2 + 5 ⎟⎠ 3. Summary for Rational Functions If N 0 = D 0 à The limit lim If N 0 < D 0 à The limit lim If N 0 > D 0 à The limit lim x→ ±∞ x→ ±∞ x→ ±∞ ( ) = the ratio of the leading coefficients of P & Q. ( ) = 0. ( ) → ±∞ P (x ) Q(x) P (x ) Q(x) P (x ) Q(x) All algebraic work must be shown when calculating limits as x→±∞!