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1. Explain the terms, left limit and right limit? 2. Evaluate (a) \( \lim _{x \rightarrow 0} \frac{x^{3}-1}{x+2} \) (b) \( \lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1} \) (c) \( \lim _{x \rightarrow 1} \frac{x^{2}+4 x-3}{3 x^{2}+2 x-5} \) (d) \( \lim _{x \rightarrow 1 \infty} \frac{x^{5}-3 x^{2}+7}{2 x^{5}+7 x-3} \) 3. What is the geometrical meaning of differentiation? 4. Differentiate the following functions using the first principle. (a) \( y=x^{2}+3 x-4 \) (b) \( y=\frac{3 x-4}{2 x+1} \) (c) \( y=\sqrt{x+3} \) (d) \( y=\frac{1}{\sqrt{3}} \) 5. Differentiate the following with respect to \( x \). (a) \( y=\frac{1}{5} x^{5}+3 x^{2}+7 x+3 \) (b) \( y=\frac{1}{6 x^{5}} \) (c) \( y=\sqrt{5 x^{2}}-\frac{1}{7 x} \) (d) \( y=\frac{2 x^{2}+7 x-6}{\sqrt{x}} \) (e) \( y=\frac{(1+x)\left(1+x^{2}\right)}{x} \) (f) \( y=(\sqrt{x}+1)(\sqrt{x}-3) \) (g) \( y=\left(x^{3}-7\right)^{10} \) (h) \( y=\frac{2 x+3}{x^{2}+x+1} \)

Ask by Gross Rose. in Zambia
Mar 15,2025

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1. **Left Limit and Right Limit**: - **Left Limit**: The value that a function approaches as \( x \) approaches a point from the left. - **Right Limit**: The value that a function approaches as \( x \) approaches a point from the right. 2. **Evaluating Limits**: - (a) \( \lim_{x \to 0} \frac{x^3 - 1}{x + 2} = -\frac{1}{2} \) - (b) \( \lim_{x \to 1} \frac{x^3 - 1}{x - 1} = 3 \) - (c) \( \lim_{x \to 1} \frac{x^2 + 4x - 3}{3x^2 + 2x - 5} \) does not exist (vertical asymptote) - (d) \( \lim_{x \to \infty} \frac{x^5 - 3x^2 + 7}{2x^5 + 7x - 3} = \frac{1}{2} \) 3. **Geometrical Meaning of Differentiation**: - Represents the slope of the tangent to the curve at a point, indicating the rate of change of the function at that point. 4. **Differentiation Using the First Principle**: - (a) \( y = x^2 + 3x - 4 \) → \( \frac{dy}{dx} = 2x + 3 \) - (b) \( y = \frac{3x - 4}{2x + 1} \) → \( \frac{dy}{dx} = \frac{11}{(2x + 1)^2} \) - (c) \( y = \sqrt{x + 3} \) → \( \frac{dy}{dx} = \frac{1}{2\sqrt{x + 3}} \) - (d) \( y = \frac{1}{\sqrt{3}} \) → \( \frac{dy}{dx} = 0 \) 5. **Differentiation of Given Functions**: - (a) \( y = \frac{1}{5}x^5 + 3x^2 + 7x + 3 \) → \( \frac{dy}{dx} = x^4 + 6x + 7 \) - (b) \( y = \frac{1}{6x} \) → \( \frac{dy}{dx} = -\frac{1}{6x^2} \) - (c) \( y = \sqrt{5x^2} - \frac{1}{7x} \) → \( \frac{dy}{dx} = \frac{5x}{\sqrt{5x^2}} + \frac{1}{7x^2} \) - (d) \( y = \frac{2x^2 + 7x - 6}{\sqrt{x}} \) → \( \frac{dy}{dx} = \frac{4x + 7}{2\sqrt{x}} - \frac{2x^2 + 7x - 6}{2x^{3/2}} \) - (e) \( y = \frac{(1 + x)(1 + x^2)}{x} \) → \( \frac{dy}{dx} = \frac{(1 + x^2)(1 + 2x) - (1 + x)(2x)}{x^2} \) - (f) \( y = (\sqrt{x} + 1)(\sqrt{x} - 3) \) → \( \frac{dy}{dx} = \frac{1}{2\sqrt{x}}(\sqrt{x} - 3) + (\sqrt{x} + 1)\frac{1}{2\sqrt{x}} \) - (g) \( y = (x^3 - 7)^{10} \) → \( \frac{dy}{dx} = 10(x^3 - 7)^9 \cdot 3x^2 \) - (h) \( y = \frac{2x + 3}{x^2 + x + 1} \) → \( \frac{dy}{dx} = \frac{(2)(x^2 + x + 1) - (2x + 3)(2x + 1)}{(x^2 + x + 1)^2} \)

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The left limit and right limit are fundamental concepts in calculus used to determine the behavior of a function as it approaches a particular point from either direction. The left limit, denoted as \(\lim_{x \to a^-} f(x)\), examines how the function behaves as \(x\) approaches \(a\) from the left side (values less than \(a\)). Conversely, the right limit, represented as \(\lim_{x \to a^+} f(x)\), observes the function as \(x\) approaches \(a\) from the right side (values greater than \(a\)). If both limits exist and are equal, the overall limit at that point can be confirmed! When it comes to real-world applications, limits are crucial in various fields, such as physics, engineering, and economics. For instance, in physics, limits help analyze motion by determining an object's velocity at a specific instant, which is historically defined as the limit of its average speed over a small interval as that interval approaches zero. In economics, limits can model scenarios like diminishing returns, helping predict changes in resource allocation. It's like applying a magnifying glass to understand the tiny details that lead to significant outcomes!

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