1. **Limits of Algebraic Functions:** - $\lim_{x \to a} c = c$ - $\lim_{x \to a} x = a$ - $\lim_{x \to a} f(x) = L$ - $\lim_{x \to a} c f(x) = c \lim_{x \to a} f(x)$ - $\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$ - $\lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$ - $\lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$ - $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ - $\lim_{x \to a} (f(x))^n = (\lim_{x \to a} f(x))^n$ - $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$ 2. **Limits of Exponential Functions:** - $\lim_{x \to a} b^x = b^a$ - $\lim_{x \to \infty} b^x = \infty$ where $b > 1$ - $\lim_{x \to -\infty} b^x = 0$ where $b > 1$ - $\lim_{x \to \infty} b^x = 0$ where $0 < b < 1$ - $\lim_{x \to -\infty} b^x = \infty$ where $0 < b < 1$ 3. **Limits of Logarithmic Functions:** - $\lim_{x \to a} \log_b x = \log_b a$ - $\lim_{x \to a} \ln x = \ln a$ - $\lim_{x \to \infty} \log_b x = \infty$ where $b > 1$ - $\lim_{x \to 0^+} \log_b x = -\infty$ where $b > 1$ - $\lim_{x \to \infty} \log_b x = -\infty$ where $0 < b < 1$ - $\lim_{x \to 0^+} \log_b x = \infty$ where $0 < b < 1$ 4. **Limits of Trigonometric Functions:** - $\lim_{x \to a} \sin x = \sin a$ - $\lim_{x \to a} \cos x = \cos a$ - $\lim_{x \to a} \tan x = \tan a$ - $\lim_{x \to a} \cot x = \cot a$ - $\lim_{x \to a} \sec x = \sec a$ - $\lim_{x \to a} \csc x = \csc a$ - $\lim_{x \to 0} \frac{\sin x}{x} = 1$ - $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ 5. **Derivative Rules of Algebraic Functions:** - $\frac{d}{dx} u^n = n u^{n-1}$ - $\frac{d}{dx} c = 0$ - $\frac{d}{dx} u = \frac{du}{dx}$ - $\frac{d}{dx} c u = c \frac{du}{dx}$ - $\frac{d}{dx} (u + v) = \frac{du}{dx} + \frac{dv}{dx}$ - $\frac{d}{dx} \sqrt{u} = \frac{1}{2 \sqrt{u}} \frac{du}{dx}$ - $\frac{d}{dx} (u \cdot v) = u \frac{dv}{dx} + v \frac{du}{dx}$ - $\frac{d}{dx} (u v w) = u w \frac{dv}{dx} + u v \frac{dw}{dx} + v w \frac{du}{dx}$ - $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$ - $\frac{d}{dx} u^n = n u^{n-1} \frac{du}{dx}$ 6. **Derivative of Logarithmic Functions:** - $\frac{d}{dx} \ln u = \frac{1}{u} \frac{du}{dx}$ - $\frac{d}{dx} \log_a u = \frac{1}{u \ln a} \frac{du}{dx}$ 7. **Derivative of Exponential Functions:** - $\frac{d}{dx} e^u = e^u \frac{du}{dx}$ 8. **Derivative of General Exponential Functions:** - $\frac{d}{dx} a^u = a^u \ln a \frac{du}{dx}$