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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) \( y=\frac{1}{\sqrt{3}} \)

Ask by Boone Reese. in Zambia
Mar 15,2025

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1. **Left Limit and Right Limit:** - **Left Limit:** The value a function approaches as the variable approaches a point from values less than that point. - **Right Limit:** The value a function approaches as the variable approaches a point from values greater than that point. 2. **(a)** \[ \lim_{x \rightarrow 0} \frac{x^3 - 1}{x + 2} = -\frac{1}{2} \] **(b)** \[ \lim_{x \rightarrow 1} \frac{x^3 - 1}{x - 1} = 3 \] **(c)** \[ \lim_{x \rightarrow 1} \frac{x^2 + 4x - 3}{3x^2 + 2x - 5} \text{ does not exist} \] **(d)** \[ \lim_{x \rightarrow \infty} \frac{x^5 - 3x^2 + 7}{2x^5 + 7x - 3} = \frac{1}{2} \] 3. **Geometrical Meaning of Differentiation:** - The derivative at a point represents the slope of the tangent line to the function's curve at that point, indicating the instantaneous rate of change of the function with respect to its variable. 4. **(a)** \[ \frac{dy}{dx} = 2x + 3 \] **(b)** \[ \frac{dy}{dx} = \frac{11}{(2x + 1)^2} \] **(c)** \[ \frac{dy}{dx} = \frac{1}{2\sqrt{x + 3}} \] **(d)** \[ \frac{dy}{dx} = 0 \]

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Beyond the Answer

Left limits and right limits are concepts in calculus that help us understand how a function behaves as it approaches a certain point from either side. The left limit, denoted as \( \lim_{x \to a^-} f(x) \), is the value that \( f(x) \) approaches as \( x \) gets closer to \( a \) from the left side. On the other hand, the right limit, represented as \( \lim_{x \to a^+} f(x) \), indicates the value that \( f(x) \) approaches as \( x \) approaches \( a \) from the right side. If both limits match, we can conclude that the limit at that point exists! When dealing with limits, particularly in your evaluations, remember the importance of factoring and simplifying. In many cases, particularly with polynomials, you can simplify expressions before taking the limit. This can help avoid common mistakes, such as directly substituting the limit point into an indeterminate form like \( \frac{0}{0} \). If you find such forms, try factoring, canceling, or applying L'Hôpital's rule as necessary.

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