2 a) Apply the limit laws, step by step, to show the following limit reduces to...

 

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2 a) Apply the limit laws, step by step, to show the following limit reduces to direct substitution { lim_(x ->1) [ (5x - 3) / (3x^2 + 5x - 2) ] b, c) Assume the following limits. Use these together with application of the limit laws to evaluate the following limits. { lim_(x -> 3) f(x) =5 } { lim_(x -> -2) f(x) = 3 %3D { lim (x -> -2) g(x) =11 } b) { lim_(x -> 3) [f(x) - x + 2]^2 c) { lim_(x -> -2) [ f(x^2 - 1)][ (f(x) + 4) / (g(x) - 3)] } QUESTION 3 continuity find the values of a and b that will make f(x) continuous for all x. { f(x) = { -x^2 - 4x + 2, x<= -1 { ax + b, { x^2 - 8x + 12, 3<= x -1 <x< 3 } a) write down the limits that makes f(x) continuous at x = -1 b) write down the limits that makes f(x) continuous at x = 3 c) solve for the values of a and b that make f(x) continuous QUESTION 4 limit definition of derivative In this problem we find the limit of a function by applying the limit method of the derivative. Consider th following function f(x). { f(x) = 2x^2 - 3x a) Set up the limit that gives the value of f(a). b) Evaluate this limit to find the derivative. QUESTION 5 Derivative - power rule, trigonometric tangent line a) Find the derivative g'(x) for the function g(x) given by the following. { g(x) = x^4 - 3x - 5/x b) Find the derivative f(x) and evaluate f(3) for the function f(x) given below. Also find the equation for the tangent line to f(x) at x=3. { f(x) = x^2 - 3x + sin(pi) x/2) } 2 a) Apply the limit laws, step by step, to show the following limit reduces to direct substitution { lim_(x ->1) [ (5x - 3) / (3x^2 + 5x - 2) ] b, c) Assume the following limits. Use these together with application of the limit laws to evaluate the following limits. { lim_(x -> 3) f(x) =5 } { lim_(x -> -2) f(x) = 3 %3D { lim (x -> -2) g(x) =11 } b) { lim_(x -> 3) [f(x) - x + 2]^2 c) { lim_(x -> -2) [ f(x^2 - 1)][ (f(x) + 4) / (g(x) - 3)] } QUESTION 3 continuity find the values of a and b that will make f(x) continuous for all x. { f(x) = { -x^2 - 4x + 2, x<= -1 { ax + b, { x^2 - 8x + 12, 3<= x -1 <x< 3 } a) write down the limits that makes f(x) continuous at x = -1 b) write down the limits that makes f(x) continuous at x = 3 c) solve for the values of a and b that make f(x) continuous QUESTION 4 limit definition of derivative In this problem we find the limit of a function by applying the limit method of the derivative. Consider th following function f(x). { f(x) = 2x^2 - 3x a) Set up the limit that gives the value of f(a). b) Evaluate this limit to find the derivative. QUESTION 5 Derivative - power rule, trigonometric tangent line a) Find the derivative g'(x) for the function g(x) given by the following. { g(x) = x^4 - 3x - 5/x b) Find the derivative f(x) and evaluate f(3) for the function f(x) given below. Also find the equation for the tangent line to f(x) at x=3. { f(x) = x^2 - 3x + sin(pi) x/2) }