Limits and Derivatives

Evaluate $\lim_{x \to 0} \frac{\sin x}{x}$.

Evaluate $\lim_{x \to 0} \frac{\tan x}{x}$.

Evaluate $\lim_{x \to 0} \frac{e^x - 1}{x}$.

Evaluate $\lim_{x \to 0} \frac{\log(1+x)}{x}$.

What is the derivative of $f(x) = x^n$ with respect to $x$?

What is the derivative of $\sin x$ with respect to $x$?

What is the derivative of a constant function $f(x) = c$?

Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$.

Evaluate $\lim_{x \to 0} \frac{\cos x - 1}{x}$.

What is the derivative of $a^x$ (where $a > 0, a \neq 1$) with respect to $x$?

If $f(x) = u(x) \cdot v(x)$, what is the derivative rule for the product of two functions?

If $f(x) = \frac{u(x)}{v(x)}$, what is the derivative rule for the quotient of two functions (where $v(x) \neq 0$)?

What is the derivative of $f(x) = 3x^2 + 5x - 7$ at $x = 1$?

Evaluate $\lim_{x \to 0} \frac{1 - \cos x}{x}$.

Find the derivative of $f(x) = \sin x$ with respect to $x$ using first principles.

Evaluate $\lim_{x \to 2} \frac{x^3 - 8}{x - 2}$.

Find the derivative of $f(x) = \cos x$.

Evaluate $\lim_{x \to 0} \frac{\log(1 + x)}{x}$.

Find the derivative of $f(x) = 3x^2 + 5x - 7$.

Evaluate $\lim_{x \to 1} \frac{x^n - 1}{x - 1}$.

Evaluate $\lim_{x \to 0} \frac{\sin(ax)}{x}$.

Find the derivative of $f(x) = \log x$ with respect to $x$.

Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.

Evaluate $\lim_{x \to 0} \frac{\cos x}{x}$.

What is the derivative of $\frac{u}{v}$ with respect to $x$ (Quotient Rule)?

Evaluate $\lim_{x \to 0} \frac{e^{2x} - 1}{x}$.