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2. Use l'Hôpital's Rule to evaluate the following limit. a) x→∞0 d) lim f) g) h) x³-1 x-1 4x²-x-3 1-cost e) lim 5x x-0 x2 2x²+3x lim x-00x²+x+1 sin x lim x-0 tan x exte-x lim x-00 ex.

2. Use l'Hôpital's Rule to evaluate the following limit. a) x→∞0 d) lim f) g) h) x³-1 x-1 4x²-x-3 1-cost e) lim 5x x-0 x2 2x²+3x lim x-00x²+x+1 sin x lim x-0 tan x exte-x lim x-00 ex.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.6: Quadratic Functions
Problem 57E
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2. Use l'Hôpital's Rule to evaluate the following limit. a) x→∞0 d) lim f) g) h) x³-1 x-1 4x²-x-3 1-cost e) lim 5x x-0 x2 2x²+3x lim x-00x²+x+1 sin x lim x-0 tan x exte-x lim x-00 ex.
Transcribed Image Text:2. Use l'Hôpital's Rule to evaluate the following limit. a) x→∞0 d) lim f) g) h) x³-1 x-1 4x²-x-3 1-cost e) lim 5x x-0 x2 2x²+3x lim x-00x²+x+1 sin x lim x-0 tan x exte-x lim x-00 ex.
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Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by f(x) at x = 0 is: ƒ'(0) f(x) = f(0) + -x + 1! f"(0) f"" (0) -x² +: 3! 2! And the Taylor series generated by f(x) at x = a is: f(x) = f(a) + 1 == a = 2 ) f'(a) 1! c) f(x) = f"(a) 2! 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b) f(x) = lnx, a = 1 π d) f(x) = sinx, a = = 4 e) f(x) = √x, a = 4 -x³ + (x − a) + -(x − a)² + · (x − a)³ + · f""'(a) 3!
Transcribed Image Text:Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by f(x) at x = 0 is: ƒ'(0) f(x) = f(0) + -x + 1! f"(0) f"" (0) -x² +: 3! 2! And the Taylor series generated by f(x) at x = a is: f(x) = f(a) + 1 == a = 2 ) f'(a) 1! c) f(x) = f"(a) 2! 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b) f(x) = lnx, a = 1 π d) f(x) = sinx, a = = 4 e) f(x) = √x, a = 4 -x³ + (x − a) + -(x − a)² + · (x − a)³ + · f""'(a) 3!
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Follow-up Question
Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by f(x) at x = 0 is: ƒ'(0) f(x) = f(0) + -x + 1! f"(0) f"" (0) -x² +: 3! 2! And the Taylor series generated by f(x) at x = a is: f(x) = f(a) + 1 == a = 2 ) f'(a) 1! c) f(x) = f"(a) 2! 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b) f(x) = lnx, a = 1 π d) f(x) = sinx, a = = 4 e) f(x) = √x, a = 4 -x³ + (x − a) + -(x − a)² + · (x − a)³ + · f""'(a) 3!
Transcribed Image Text:Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by f(x) at x = 0 is: ƒ'(0) f(x) = f(0) + -x + 1! f"(0) f"" (0) -x² +: 3! 2! And the Taylor series generated by f(x) at x = a is: f(x) = f(a) + 1 == a = 2 ) f'(a) 1! c) f(x) = f"(a) 2! 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b) f(x) = lnx, a = 1 π d) f(x) = sinx, a = = 4 e) f(x) = √x, a = 4 -x³ + (x − a) + -(x − a)² + · (x − a)³ + · f""'(a) 3!
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Follow-up Question
Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by f(x) at x = 0 is: ƒ'(0) f(x) = f(0) + -x + 1! f"(0) f"" (0) -x² +: 3! 2! And the Taylor series generated by f(x) at x = a is: f(x) = f(a) + 1 == a = 2 ) f'(a) 1! c) f(x) = f"(a) 2! 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b) f(x) = lnx, a = 1 π d) f(x) = sinx, a = = 4 e) f(x) = √x, a = 4 -x³ + (x − a) + -(x − a)² + · (x − a)³ + · f""'(a) 3!
Transcribed Image Text:Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by f(x) at x = 0 is: ƒ'(0) f(x) = f(0) + -x + 1! f"(0) f"" (0) -x² +: 3! 2! And the Taylor series generated by f(x) at x = a is: f(x) = f(a) + 1 == a = 2 ) f'(a) 1! c) f(x) = f"(a) 2! 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b) f(x) = lnx, a = 1 π d) f(x) = sinx, a = = 4 e) f(x) = √x, a = 4 -x³ + (x − a) + -(x − a)² + · (x − a)³ + · f""'(a) 3!
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Follow-up Question
Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by f(x) at x = 0 is: ƒ'(0) f(x) = f(0) + -x + 1! f"(0) f"" (0) -x² +: 3! 2! And the Taylor series generated by f(x) at x = a is: f(x) = f(a) + 1 == a = 2 ) f'(a) 1! c) f(x) = f"(a) 2! 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b) f(x) = lnx, a = 1 π d) f(x) = sinx, a = = 4 e) f(x) = √x, a = 4 -x³ + (x − a) + -(x − a)² + · (x − a)³ + · f""'(a) 3!
Transcribed Image Text:Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by f(x) at x = 0 is: ƒ'(0) f(x) = f(0) + -x + 1! f"(0) f"" (0) -x² +: 3! 2! And the Taylor series generated by f(x) at x = a is: f(x) = f(a) + 1 == a = 2 ) f'(a) 1! c) f(x) = f"(a) 2! 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b) f(x) = lnx, a = 1 π d) f(x) = sinx, a = = 4 e) f(x) = √x, a = 4 -x³ + (x − a) + -(x − a)² + · (x − a)³ + · f""'(a) 3!
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