Preface

This is the note of the MIT Course 18.01.1 Calculus. For more details, refer to MIT 18.x Catalog, and it can also be found at MIT Open Learning Library.

Unit 0: Limits

Definition of One-Sided Limit

Suppose $f(x)$ gets really close to $R$ for values of $x$ that get really close to and are not equal to $a$ from the right. Then we say $R$ is the right-hand limit of the function $f(x)$ as $x$ approaches $a$ from the right.

We note $$ f(x)\to R\ as\ x\to a^+\\ or\\ \lim_{x\to a^+}f(x)=R $$ If $f(x)$ gets really close to $L$ for values of $x$ that get really close to and are not equal to $a$ from the left, we say that $L$ is the left-hand limit of the function $f(x)$ as $x$ approaches $a$ from the left.

We note $$ f(x)\to L\ as\ x\to a^-\\ or\\ \lim_{x\to a^-}f(x)=L $$

Definition of the Limit

If a function $f(x)$ approaches some value $L$ as $x$ approaches $a$ from both the right and the left, then the limit of $f(x)$ exists and equals $L$.

Write the limit in symbols, if $$ \lim_{x\to a^+}f(x)=\lim_{x\to a^-}f(x)=L $$ then $$ \lim_{x\to a}f(x) = L $$ In other way, $$ f(x)\to L \quad as\quad x\to a $$ The important thing is that $x$ is approaching $a$ but does not equal $a$.

Here is the formal definition of the limit:

The statement $\lim_{x\to a}f(x)=L$ is defined as:

For all $\varepsilon>0$, there exists some $\delta>0$ such that if $0<|x-a|<\delta$, then $|f(x)-L|<\varepsilon$.

The Limit Laws

Suppose $\lim_{x\to a}f(x)=L,\quad \lim_{x\to a}g(x)=M$.

Then here are some Limit Laws: $$ \begin{aligned} \text{Addition:}\quad &\lim_{x\to a}[f(x)+g(x)]&=&\quad L+M\\ \text{Subtraction:}\quad &\lim_{x\to a}[f(x)-g(x)]&=&\quad L-M\\ \text{Multiplication:}\quad &\lim_{x\to a}[f(x)\cdot g(x)]&=&\quad L\cdot M \end{aligned} $$ It is noteworthy that we only have part of the Limit Law for Divison:

If $\lim_{x\to a}f(x)=L$ and $\lim_{x\to a}g(x)=M$, then:

  • If $M\ne 0$, then $\lim_{x\to a}\frac {f(x)}{g(x)}=\frac L M$
  • If $M=0$, then $\lim_{x\to a}\frac{f(x)}{g(x)}$ does not exist
  • If $M=0$ and $L= 0$, then $\lim_{x\to a}\frac{f(x)}{g(x)}$ might exist or not exist. It is necessary to do more work to determine whether or not its limit exists, and what it is if it does exist.

Definition of Continuous at a Point

We say that a function $f$ is continuous at a point $x=a$ if $$ \lim_{x\to a}f(x)=f(a) $$ In particular, if either $f(a)$ or $\lim_{x\to a}f(x)$ fails to exist, then $f$ is discontinuous at $a$.

We say that a function $f$ is right-continuous at a point $x=a$ if $\lim_{x\to a^+}f(x)=f(a)$.

We say that a function $f$ is left-continuous at a point $x=a$ if $\lim_{x\to a^-}f(x)=f(a)$.

We say that a function $f$ has a jump discontinuity at $x=a$ if both of the left-hand limit $\lim_{x\to a^-}f(x)$ and the right-hand limit $\lim_{x\to a^+}f(x)$ exist but they are not equal.

We say that a function $f$ has a removable discontinuity at $x=a$ if the overall limit $\lim_{x\to a}f(x)$ exists, but the $\lim_{x\to a}f(x)\ne f(a)$.

Basic Continuous Functions

The following functions are continuous at all real numbers:

  • all polynomials
  • $\sqrt[3]{x}$
  • $|x|$
  • $\cos(x)$ and $\sin(x)$
  • exponential functions $a^x$ which has $a>0$.

The following functions are continuous at the specified values of $x$:

  • $\sqrt{x}, \forall x>0$
  • $\tan(x), \forall x\ \text{is defined}$
  • logarithmic functions $\log_a x$, where $a\in(0,\infty)$ with $a\neq 1$ and $x\in(0,\infty)$.

Intermediate Value Theorem

If the function $f$ is continuous on the interval $[a,b]$, and $M$ lies between the values of $f(a)$ and $f(b)$, then there is at least one point $c$ between $a$ and $b$ such that $f(c)=M$.

In other words, a function $f$ is continuous on a closed interval $[a,b]$ if it is right-continuous at $a$, left-continuous at $b$, and continuous at all points between $a$ and $b$.