Define limit of a function. List the methods for evaluating the limits of a function. Evaluate the limits of the following functions. Itemize four (4) techniques for evaluating differentiation together with their formula.

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Asked on September 8, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-09-08T15:05:38.049Z

Question One: 1a) Define limit of a function A limit of a function describes the value that a function "approaches" as the input (or independent variable) approaches a certain value. It is a fundamental concept in calculus for defining continuity, derivatives, and integrals. 1b) List the methods for evaluating the limits of a function • Direct Substitution • Factoring and Cancellation • Rationalization • L'Hôpital's Rule • Squeeze Theorem • Using known limits (e.g., trigonometric limits) 1c) Evaluate the limits of the following functions i. _x -1 (x^3 + 4x - 2) Step 1: Substitute x = -1 into the expression. _x -1 (x^3 + 4x - 2) = (-1)^3 + 4(-1) - 2 Step 2: Simplify the expression. = -1 - 4 - 2 = -7 The limit is -7. ii. _x 2 (x^2 - 4)/(x - 2) Step 1: Factor the numerator. _x 2 ((x - 2)(x + 2))/(x - 2) Step 2: Cancel out the common term (x - 2). _x 2 (x + 2) Step 3: Substitute x = 2 into the simplified expression. = 2 + 2 = 4 The limit is 4. iii. _x 0 (x^2 + 1)/(x^2 - x) Step 1: Substitute x = 0 into the expression. ((0)^2 + 1)/((0)^2 - 0) = (1)/(0) Step 2: Since the denominator approaches 0 and the numerator approaches 1, the limit does not exist. The limit is Does\ not\ exist. Question Two: 2a) Itemize four (4) techniques for evaluating differentiation together with their formula • Power Rule: If y = x^n, then (dy)/(dx) = nx^n-1. • Product Rule: If y = u(x)v(x), then (dy)/(dx) = u'(x)v(x) + u(x)v'(x). • Quotient Rule: If y = (u(x))/(v(x)), then (dy)/(dx) = (u'(x)v(x) - u(x)v'(x))/([v(x)]^2). • Chain Rule: If y = f(g(x)), then (dy)/(dx) = f'(g(x)) · g'(x). 2b) if y = x^2, differentiate using the first principle The first principle definition of the derivative is (dy)/(dx) = _h 0 (f(x+h) - f(x))/(h). Step 1: Substitute f(x) = x^2 into the formula. (dy)/(dx) = _h 0 ((x+h)^2 - x^2)/(h) Step 2: Expand (x+h)^2. = _h 0 (x^2 + 2xh + h^2 - x^2)/(h) Step 3: Simplify the numerator. = _h 0 (2xh + h^2)/(h) Step 4: Factor out h from the numerator. = _h 0 (h(2x + h))/(h) Step 5: Cancel h and substitute h=0. = _h 0 (2x + h) = 2x + 0 = 2x The derivative is 2x. 2c) Find the derivatives of the following with respect to x using appropriate techniques of differentiation i. y = (x^2 + 2)^5 Step 1: Apply the chain rule. Let u = x^2 + 2, so y = u^5. Then (dy)/(du) = 5u^4 and (du)/(dx) = 2x. Step 2: Multiply (dy)/(du) by (du)/(dx). (dy)/(dx) = 5(x^2 + 2)^4 · (2x) Step 3: Simplify the expression. (dy)/(dx) = 10x(x^2 + 2)^4 The derivative is 10x(x^2 + 2)^4. ii. y = ( x)/( x) Step 1: Recognize that y = x. Step 2: Differentiate y = x. (dy)/(dx) = ^2 x Alternatively, using the quotient rule: Step 1: Let u = x and v = x. Then u' = x and v' = - x. Step 2: Apply the quotient rule (dy)/(dx) = (u'v - uv')/(v^2). (dy)/(dx) = (( x)( x) - ( x)(- x))/(( x)^2) Step 3: Simplify the numerator. = (^2 x + ^2 x)/(^2 x) Step 4: Use the identity ^2 x + ^2 x = 1. = (1)/(^2 x) Step 5: Rewrite in terms of x. = ^2 x The derivative is ^2 x. Question Three: 3a) Differentiate y = e^x _e x using product rule. Step 1: Identify u = e^x and v = _e x. Step 2: Find the derivatives u' and v'. u' = (d)/(dx)(e^x) = e^x v' = (d)/(dx)(_e x) = (1)/(x) Step 3: Apply the product rule: (dy)/(dx) = u'v + uv'. (dy)/(dx) = (e^x)(_e x) + (e^x)((1)/(x)) Step 4: Factor out e^x. (dy)/(dx) = e^x (_e x + (1)/(x)) The derivative is e^x (_e x + (1)/(x)). 3b) Find (dy)/(dx), if x^2 + y^2 + xy^3 = 3 at the given point (1, 1). Step 1: Differentiate both sides of the equation with respect to x. Remember to use the chain rule for terms involving y and the product rule for xy^3. (d)/(dx)(x^2) + (d)/(dx)(y^2) + (d)/(dx)(xy^3) = (d)/(dx)(3) 2x + 2y (dy)/(dx) + (1 · y^3 + x · 3y^2 (dy)/(dx)) = 0 Step 2: Group terms with (dy)/(dx). 2x + y^3 + (2y + 3xy^2) (dy)/(dx) = 0 Step 3: Isolate (dy)/(dx). (2y + 3xy^2) (dy)/(dx) = -2x - y^3 (dy)/(dx) = (-2x - y^3)/(2y + 3xy^2) Step 4: Substitute the point (x, y) = (1, 1) into the expression for (dy)/(dx). (dy)/(dx) = (-2(1) - (1)^3)/(2(1) + 3(1)(1)^2) = (-2 - 1)/(2 + 3) = (-3)/(5) At the point (1, 1), (dy)/(dx) is -(3)/(5). 3c) if y = (t^2)/(1+t), x = (t)/(1+t), then find (dy)/(dx). Step 1: Find (dy)/(dt) using the quotient rule. Let u = t^2, v = 1+t. Then u' = 2t, v' = 1. (dy)/(dt) = (u'v - uv')/(v^2) = (2t(1+t) - t^2(1))/((1+t)^2) = (2t + 2t^2 - t^2)/((1+t)^2) = (t^2 + 2t)/((1+t)^2) Step 2: Find (dx)/(dt) using the quotient rule. Let u = t, v = 1+t. Then u' = 1, v' = 1. (dx)/(dt) = (u'v - uv')/(v^2) = (1(1+t) - t(1))/((1+t)^2) = (1+t - t)/((1+t)^2) = (1)/((1+t)^2) Step 3: Use the chain rule for parametric equations: (dy)/(dx) = (dy/dt)/(dx/dt). (dy)/(dx) = (t^2 + 2t)/((1+t)^2)(1)/((1+t)^2) Step 4: Simplify the expression. (dy)/(dx) = (t^2 + 2t) · ((1+t)^2)/(1) = t^2 + 2t The derivative is t^2 + 2t. Send me the next one 📸

Accepted Answer · 2026-09-08T15:05:38.049Z

Question One: 1a) Define limit of a function A limit of a function describes the value that a function "approaches" as the input (or independent variable) approaches a certain value. It is a fundamental concept in calculus for defining continuity, derivatives, and integrals. 1b) List the methods for evaluating the limits of a function • Direct Substitution • Factoring and Cancellation • Rationalization • L'Hôpital's Rule • Squeeze Theorem • Using known limits (e.g., trigonometric limits) 1c) Evaluate the limits of the following functions i. _x -1 (x^3 + 4x - 2) Step 1: Substitute x = -1 into the expression. _x -1 (x^3 + 4x - 2) = (-1)^3 + 4(-1) - 2 Step 2: Simplify the expression. = -1 - 4 - 2 = -7 The limit is -7. ii. _x 2 (x^2 - 4)/(x - 2) Step 1: Factor the numerator. _x 2 ((x - 2)(x + 2))/(x - 2) Step 2: Cancel out the common term (x - 2). _x 2 (x + 2) Step 3: Substitute x = 2 into the simplified expression. = 2 + 2 = 4 The limit is 4. iii. _x 0 (x^2 + 1)/(x^2 - x) Step 1: Substitute x = 0 into the expression. ((0)^2 + 1)/((0)^2 - 0) = (1)/(0) Step 2: Since the denominator approaches 0 and the numerator approaches 1, the limit does not exist. The limit is Does\ not\ exist. Question Two: 2a) Itemize four (4) techniques for evaluating differentiation together with their formula • Power Rule: If y = x^n, then (dy)/(dx) = nx^n-1. • Product Rule: If y = u(x)v(x), then (dy)/(dx) = u'(x)v(x) + u(x)v'(x). • Quotient Rule: If y = (u(x))/(v(x)), then (dy)/(dx) = (u'(x)v(x) - u(x)v'(x))/([v(x)]^2). • Chain Rule: If y = f(g(x)), then (dy)/(dx) = f'(g(x)) · g'(x). 2b) if y = x^2, differentiate using the first principle The first principle definition of the derivative is (dy)/(dx) = _h 0 (f(x+h) - f(x))/(h). Step 1: Substitute f(x) = x^2 into the formula. (dy)/(dx) = _h 0 ((x+h)^2 - x^2)/(h) Step 2: Expand (x+h)^2. = _h 0 (x^2 + 2xh + h^2 - x^2)/(h) Step 3: Simplify the numerator. = _h 0 (2xh + h^2)/(h) Step 4: Factor out h from the numerator. = _h 0 (h(2x + h))/(h) Step 5: Cancel h and substitute h=0. = _h 0 (2x + h) = 2x + 0 = 2x The derivative is 2x. 2c) Find the derivatives of the following with respect to x using appropriate techniques of differentiation i. y = (x^2 + 2)^5 Step 1: Apply the chain rule. Let u = x^2 + 2, so y = u^5. Then (dy)/(du) = 5u^4 and (du)/(dx) = 2x. Step 2: Multiply (dy)/(du) by (du)/(dx). (dy)/(dx) = 5(x^2 + 2)^4 · (2x) Step 3: Simplify the expression. (dy)/(dx) = 10x(x^2 + 2)^4 The derivative is 10x(x^2 + 2)^4. ii. y = ( x)/( x) Step 1: Recognize that y = x. Step 2: Differentiate y = x. (dy)/(dx) = ^2 x Alternatively, using the quotient rule: Step 1: Let u = x and v = x. Then u' = x and v' = - x. Step 2: Apply the quotient rule (dy)/(dx) = (u'v - uv')/(v^2). (dy)/(dx) = (( x)( x) - ( x)(- x))/(( x)^2) Step 3: Simplify the numerator. = (^2 x + ^2 x)/(^2 x) Step 4: Use the identity ^2 x + ^2 x = 1. = (1)/(^2 x) Step 5: Rewrite in terms of x. = ^2 x The derivative is ^2 x. Question Three: 3a) Differentiate y = e^x _e x using product rule. Step 1: Identify u = e^x and v = _e x. Step 2: Find the derivatives u' and v'. u' = (d)/(dx)(e^x) = e^x v' = (d)/(dx)(_e x) = (1)/(x) Step 3: Apply the product rule: (dy)/(dx) = u'v + uv'. (dy)/(dx) = (e^x)(_e x) + (e^x)((1)/(x)) Step 4: Factor out e^x. (dy)/(dx) = e^x (_e x + (1)/(x)) The derivative is e^x (_e x + (1)/(x)). 3b) Find (dy)/(dx), if x^2 + y^2 + xy^3 = 3 at the given point (1, 1). Step 1: Differentiate both sides of the equation with respect to x. Remember to use the chain rule for terms involving y and the product rule for xy^3. (d)/(dx)(x^2) + (d)/(dx)(y^2) + (d)/(dx)(xy^3) = (d)/(dx)(3) 2x + 2y (dy)/(dx) + (1 · y^3 + x · 3y^2 (dy)/(dx)) = 0 Step 2: Group terms with (dy)/(dx). 2x + y^3 + (2y + 3xy^2) (dy)/(dx) = 0 Step 3: Isolate (dy)/(dx). (2y + 3xy^2) (dy)/(dx) = -2x - y^3 (dy)/(dx) = (-2x - y^3)/(2y + 3xy^2) Step 4: Substitute the point (x, y) = (1, 1) into the expression for (dy)/(dx). (dy)/(dx) = (-2(1) - (1)^3)/(2(1) + 3(1)(1)^2) = (-2 - 1)/(2 + 3) = (-3)/(5) At the point (1, 1), (dy)/(dx) is -(3)/(5). 3c) if y = (t^2)/(1+t), x = (t)/(1+t), then find (dy)/(dx). Step 1: Find (dy)/(dt) using the quotient rule. Let u = t^2, v = 1+t. Then u' = 2t, v' = 1. (dy)/(dt) = (u'v - uv')/(v^2) = (2t(1+t) - t^2(1))/((1+t)^2) = (2t + 2t^2 - t^2)/((1+t)^2) = (t^2 + 2t)/((1+t)^2) Step 2: Find (dx)/(dt) using the quotient rule. Let u = t, v = 1+t. Then u' = 1, v' = 1. (dx)/(dt) = (u'v - uv')/(v^2) = (1(1+t) - t(1))/((1+t)^2) = (1+t - t)/((1+t)^2) = (1)/((1+t)^2) Step 3: Use the chain rule for parametric equations: (dy)/(dx) = (dy/dt)/(dx/dt). (dy)/(dx) = (t^2 + 2t)/((1+t)^2)(1)/((1+t)^2) Step 4: Simplify the expression. (dy)/(dx) = (t^2 + 2t) · ((1+t)^2)/(1) = t^2 + 2t The derivative is t^2 + 2t. Send me the next one 📸

Define limit of a function. List the methods for evaluating the limits of a function. Evaluate the limits of the following functions. Itemize four (4) techniques for evaluating differentiation together with their formula.

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Define limit of a function. List the methods for evaluating the limits of a function. Evaluate the limits of the following functions. Itemize four (4) techniques for evaluating differentiation together with their formula.

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