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In Class 11, limits and derivatives are introduced as fundamental concepts in calculus, focusing on understanding how functions behave and how they change. Here’s an overview of these topics:

1. **Limits**: A limit is the value that a function approaches as the input approaches a certain value. Formally, the limit of a function $$f(x)$$ as $$x$$ approaches $$a$$ is denoted by $$\lim_{x \to a} f(x)$$. Some key concepts related to limits include:
– One-sided limits: Limits can be approached from the left ($$\lim_{x \to a^-}$$) or from the right ($$\lim_{x \to a^+}$$).
– Limit laws: Rules that allow for the computation of limits based on the limits of simpler functions.
– Continuity: A function is continuous at a point if the limit of the function exists at that point and is equal to the value of the function at that point.

2. **Derivatives**: A derivative measures the rate at which a function changes with respect to its input. It represents the slope of the tangent line to the graph of the function at a given point. Formally, the derivative of a function $$f(x)$$ with respect to $$x$$ is denoted by $$f'(x)$$ or $$\frac{df}{dx}$$. Key concepts related to derivatives include:
– Derivative as a limit: The derivative of $$f(x)$$ at $$x = a$$ is defined as $$\lim_{h \to 0} \frac{f(a + h) – f(a)}{h}$$.
– Differentiability: A function is differentiable at a point if the derivative exists at that point.
– Rules of differentiation: Formulas for finding the derivatives of common functions, including the power rule, product rule, quotient rule, and chain rule.

3. **Applications of Derivatives**: Derivatives have many applications in mathematics and science, including:
– Calculating rates of change: Derivatives can be used to find rates of change of quantities.
– Optimization: Derivatives can help find maximum and minimum values of functions.
– Curve sketching: Derivatives provide information about the behavior of functions, which can be used to sketch their graphs.

Understanding limits and derivatives is essential for further studies in calculus and mathematics, as they form the basis for more advanced topics such as integrals and differential equations.