# Limits Continuity and Differentiability

### Limits:

**Definition:** A limit represents the value that a function approaches as the input approaches a certain value.

**Notation:** The limit of a function $f(x)$ as $x$approaches $a$ is denoted by ${\mathrm{lim}}_{x\to a}f(x)$.

**Properties:**

**Limit Laws:**- Sum/Difference Law: ${\mathrm{lim}}_{x\to a}[f(x)\pm g(x)]={\mathrm{lim}}_{x\to a}f(x)\pm {\mathrm{lim}}_{x\to a}g(x)$
- Product Law: ${\mathrm{lim}}_{x\to a}[f(x)\cdot g(x)]={\mathrm{lim}}_{x\to a}f(x)\cdot {\mathrm{lim}}_{x\to a}g(x)$
- Quotient Law: ${\mathrm{lim}}_{x\to a}\frac{f(x)}{g(x)}=\frac{{\mathrm{lim}}_{x\to a}f(x)}{{\mathrm{lim}}_{x\to a}g(x)}$
- Power Law: ${\mathrm{lim}}_{x\to a}[f(x){]}^{n}=[{\mathrm{lim}}_{x\to a}f(x){]}^{n}$ (where $n$ is a constant)

**Types of Limits:**

**One-sided limits:**Limits calculated from one direction (left or right) of a specific point.**Infinite limits:**When the limit of a function approaches infinity or negative infinity.

### Continuity:

**Definition:** A function is continuous at a point if the function is defined at that point, the limit of the function exists at that point, and the limit equals the value of the function at that point.

**Types of Discontinuities:**

**Removable Discontinuity:**A hole in the graph that can be filled by redefining the function at that point.**Jump Discontinuity:**The function "jumps" from one value to another at a specific point.**Infinite Discontinuity:**Asymptotic behavior where the function approaches infinity or negative infinity at a certain point.

**Continuity Theorems:**

**Intermediate Value Theorem:**If a function $f(x)$ is continuous on the interval $[a,b]$ and $f(a)$ and $f(b)$ have different signs, then there exists at least one value $c$in the interval $[a,b]$such that $f(c)=0$.**Extreme Value Theorem:**If a function $f(x)$ is continuous on a closed interval $[a,b]$, then it has both a maximum and minimum value on that interval.

### Differentiability:

**Definition:** A function is differentiable at a point if the derivative at that point exists.

**Differentiability Conditions:**

**Differentiable implies Continuous:**If a function is differentiable at a point, it must be continuous at that point.**Differentiation Rules:**- Power Rule: $\frac{d}{dx}[{x}^{n}]=n{x}^{n-1}$
- Sum/Difference Rule: $\frac{d}{dx}[f(x)\pm g(x)]=\frac{d}{dx}[f(x)]\pm \frac{d}{dx}[g(x)]$
- Product Rule: $\frac{d}{dx}[f(x)\cdot g(x)]={f}^{\mathrm{\prime}}(x)\cdot g(x)+f(x)\cdot {g}^{\mathrm{\prime}}(x)$
- Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{{f}^{\mathrm{\prime}}(x)\cdot g(x)-f(x)\cdot {g}^{\mathrm{\prime}}(x)}{[g(x){]}^{2}}$

**Differentiability Theorems:**

**Rolle's Theorem:**If a function $f(x)$ is continuous on the closed interval $[a,b]$, differentiable on the open interval $(a,b)$, and $f(a)=f(b)$, then there exists at least one $c$ in the interval $(a,b)$such that ${f}^{\mathrm{\prime}}(c)=0$.**Mean Value Theorem:**If a function $f(x)$ is continuous on the closed interval $[a,b]$, differentiable on the open interval $(a,b)$, then there exists at least one $c$ in the interval $(a,b)$such that $\frac{f(b)-f(a)}{b-a}={f}^{\mathrm{\prime}}(c)$.