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Determine the following limits. If the value is infinite, specify c∞ or -o with enough justification for which. If it does not exist, explain why. 3x – x² + 12x – 4 lim 3x4 – x3 + 3x2 +2x – 1

Determine the following limits. If the value is infinite, specify c∞ or -o with enough justification for which. If it does not exist, explain why. 3x – x² + 12x – 4 lim 3x4 – x3 + 3x2 +2x – 1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 55E
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Please solve this using the given formula sheet, and solve Without using L'Hopital's rule. Thank you.

LIMIT LAWS Sum: lim [f(x) + g(x)] = lim f(x) + lim g(x) Difference: lim [f(x) - g(x)] = lim f(x) – lim g(x) Constant Multiple: lim [c ·f(x)] = c - lim f(x) Product: lim lim f(x) g(x) lim f(x) f(x) Quotient: lim 1-a (g(x) provided lim g(x) # 0 lim g (x) Power. lim [f(x)]" = | lim f(x) n/m Fractional Power. lim [f(x)]" = | lim n/m provided f(x) 2 0 for x near a if m is even and n Im is reduced to lowest term IMPORTANT LIMITS SQUEEZE THEOREM sin(x) lim Suppose f, g, and h are functions so that f(x) < g(x) < h(x) for all x close to a, where lim f(x) = L and lim h(x) = L, then = 1 X-0 cos(x) – 1 lim = 0 lim g(x) = L X-0 PRECISE LIMIT DEFINITIONS CONTINUITY Finite Limit: For any e > 0, there is some ô > 0 so that if 0 < ]x – a| < 8, then |f(x) – L| < e. A function fis continuous at a point x = a VE C"(a)] if it satisfies the following: 1. a is in the domain of f Positive Infinite Limit: For any N > 0, there is some 8 > 0 so that if 0 < ]x – a|< 8, then f(x) > N. 2. lim f(x) exists 3. lim f(x) = f(a) Negative Infinite Limit: For any N > 0, there is some ô >0 so that if 0 < [x – a|< ô, then f (x) < – N. DISCONTINUITIES Limit at Positive Infinity For any e > 0, there is some N > 0 so that if x > N, then |f(x) – L|<e. Limit at Negative Infinity For any e > 0, there is some N > 0 so that if x<-N, then [f(x) – L|<e. A removable discontinuity occurs when condition 2. is satisfied but 1. or 3. is not. A jump discontinuity occurs when the one- sided limits are finite but disagree. An infinite discontinuity occurs when at least one of the one-sided limits are infinite. INTERMEDIATE VALUE THEOREM Suppose fis continuous on the interval [a,b] and L is a number betweenf(a) and f(b). Then there is at least one number c in the interval |(a,b) which satisfies f(c) = L.
Transcribed Image Text:LIMIT LAWS Sum: lim [f(x) + g(x)] = lim f(x) + lim g(x) Difference: lim [f(x) - g(x)] = lim f(x) – lim g(x) Constant Multiple: lim [c ·f(x)] = c - lim f(x) Product: lim lim f(x) g(x) lim f(x) f(x) Quotient: lim 1-a (g(x) provided lim g(x) # 0 lim g (x) Power. lim [f(x)]" = | lim f(x) n/m Fractional Power. lim [f(x)]" = | lim n/m provided f(x) 2 0 for x near a if m is even and n Im is reduced to lowest term IMPORTANT LIMITS SQUEEZE THEOREM sin(x) lim Suppose f, g, and h are functions so that f(x) < g(x) < h(x) for all x close to a, where lim f(x) = L and lim h(x) = L, then = 1 X-0 cos(x) – 1 lim = 0 lim g(x) = L X-0 PRECISE LIMIT DEFINITIONS CONTINUITY Finite Limit: For any e > 0, there is some ô > 0 so that if 0 < ]x – a| < 8, then |f(x) – L| < e. A function fis continuous at a point x = a VE C"(a)] if it satisfies the following: 1. a is in the domain of f Positive Infinite Limit: For any N > 0, there is some 8 > 0 so that if 0 < ]x – a|< 8, then f(x) > N. 2. lim f(x) exists 3. lim f(x) = f(a) Negative Infinite Limit: For any N > 0, there is some ô >0 so that if 0 < [x – a|< ô, then f (x) < – N. DISCONTINUITIES Limit at Positive Infinity For any e > 0, there is some N > 0 so that if x > N, then |f(x) – L|<e. Limit at Negative Infinity For any e > 0, there is some N > 0 so that if x<-N, then [f(x) – L|<e. A removable discontinuity occurs when condition 2. is satisfied but 1. or 3. is not. A jump discontinuity occurs when the one- sided limits are finite but disagree. An infinite discontinuity occurs when at least one of the one-sided limits are infinite. INTERMEDIATE VALUE THEOREM Suppose fis continuous on the interval [a,b] and L is a number betweenf(a) and f(b). Then there is at least one number c in the interval |(a,b) which satisfies f(c) = L.
Determine the following limits. If the value is infinite, specify co or -o with enough justification for which. If it does not exist, explain why. 3x³ – x² + 12x – 4 lim 3x4 – x3 +3x2 +2x – 1
Transcribed Image Text:Determine the following limits. If the value is infinite, specify co or -o with enough justification for which. If it does not exist, explain why. 3x³ – x² + 12x – 4 lim 3x4 – x3 +3x2 +2x – 1
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