Key Points

The limit of a function $f(x)$ as $x$ approaches a point $a$, denoted $\lim_{x \rightarrow a} f(x) = l$, is the value that $f(x)$ gets arbitrarily close to as $x$ approaches $a$.

The Left-Hand Limit (LHL) is $\lim_{x \rightarrow a^{-}} f(x)$, approaching $a$ from values less than $a$. The Right-Hand Limit (RHL) is $\lim_{x \rightarrow a^{+}} f(x)$, approaching from values greater than $a$.

A limit of a function at a point $a$ exists if and only if its Left-Hand Limit and Right-Hand Limit at that point are equal and finite.

The limit of the sum or difference of two functions is the sum or difference of their individual limits: $\lim_{x \rightarrow a} [f(x) \pm g(x)] = \lim_{x \rightarrow a} f(x) \pm \lim_{x \rightarrow a} g(x)$.

The limit of a product is the product of limits. The limit of a quotient is the quotient of limits, provided the denominator's limit is non-zero: $\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)}$, where $\lim_{x \rightarrow a} g(x) \neq 0$.

For a polynomial or rational function $f(x)$, if $f(a)$ is defined, then $\lim_{x \rightarrow a} f(x) = f(a)$. If it results in the form $\frac{0}{0}$, algebraic simplification is needed.

A fundamental limit used for polynomials is $\lim_{x \rightarrow a} \frac{x^n - a^n}{x - a} = na^{n-1}$ for any rational number $n$.

One of the most important trigonometric limits is $\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$, where $x$ is measured in radians.

Another fundamental trigonometric limit is $\lim_{x \rightarrow 0} \frac{1 - \cos x}{x} = 0$.

If $f(x) \le g(x) \le h(x)$ for all $x$ near $a$, and $\lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} h(x) = L$, then $\lim_{x \rightarrow a} g(x) = L$.

The derivative of a function $f(x)$ with respect to $x$, denoted $f'(x)$, is defined by the limit $f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$. This is known as the first principle of derivatives.

The derivative of a function $f(x)$ at a point $x=a$, i.e., $f'(a)$, represents the slope of the tangent line to the graph of $y = f(x)$ at the point $(a, f(a))$.

The derivative of $f(x) = x^n$ for any positive integer $n$ is given by the formula $\frac{d}{dx}(x^n) = nx^{n-1}$.

The derivative of the sum or difference of two differentiable functions is the sum or difference of their derivatives: $\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)$.

The derivative of the product of two functions is given by the Leibnitz rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$.

The derivative of the quotient of two functions is given by $\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$, provided $g(x) \neq 0$.

The derivatives of sine and cosine functions are fundamental: $\frac{d}{dx}(\sin x) = \cos x$ and $\frac{d}{dx}(\cos x) = -\sin x$.