Limits and Continuity

Limit of a Sequence

When a sequence of values of a variable $x$ approach nearer and nearer to a fixed value $a$ in such a way that the differences $|x-a|$ becomes and remains as small as we please, the value $a$ is called the limit of the variable $x$.

A variable $x$ is said to have a limit $a$ as $x$ takes on the values $x_1, x_2, x_3, \dots, x_n$ if for every positive number $\epsilon$, however small, the value of $|x_n-a|<\epsilon$ for suitably large $n$. We write $$\lim_{n\to \infty}x_n=a$$

Limit of a Function

A function $f(x)$ is said to have a limit $b$ as $x$ approaches a value $a$ if for every positive number $\epsilon$, however small there is a value $\delta$ so that $|f(x)-b|<\epsilon$ whenever $|x-a|<\delta$. We write $$\lim_{x\to a}f(x)=b.$$

In the above diagram $$\lim_{x\to a}f(x)=a.$$

Operations with Limits

Suppose that $\lim_{x\to a}f(x)$ and $\lim_{x\to a}g(x)$ both exist and let $c$ be any constant. Then $$\lim_{x\to a}c. f(x)=c.\lim_{x\to a}f(x)$$ $$\lim_{x\to a}\left[f(x)\pm g(x)\right]=\lim_{x\to a}f(x) \pm\lim_{x\to a}g(x)$$ $$\lim_{x\to a}\left[f(x).g(x)\right]=\lim_{x\to a}f(x).\lim_{x\to a}g(x)$$ $$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\displaystyle{\lim_{x\to a}f(x)}}{\displaystyle{\lim_{x\to a}g(x)}}\text{ provided }\lim_{x\to a}g(x)\ne 0$$

Special Limits for $x\to 0$ and $x\to 1$

$\displaystyle{\lim_{x\to 0}a^x=1},\quad a>0$
$\displaystyle{\lim_{x\to 0}\left(1+x\right)^{1/x}=e}$
$\displaystyle{\lim_{x\to 0}\frac{e^x-1}{x}=1}$
$\displaystyle{\lim_{x\to 0}\frac{\sin x}{x}=1}$
$\displaystyle{\lim_{x\to 0}\frac{1-\cos x}{x}=0}$
$\displaystyle{\lim_{x\to 0}\frac{\tan x}{x}=1}$
$\displaystyle{\lim_{x\to 1}\frac{x-1}{\ln x}=1}$

Special Limits for $x\to\infty$

$\displaystyle{\lim_{x\to \infty}\left(1+\frac{y}{x}\right)^x=e^y}$
$\displaystyle{\lim_{x\to \infty}\frac{a^x-1}{x}=\ln a,\quad a>0}$
$\displaystyle{\lim_{m\to \infty}\frac{a^m}{m!}=0}$
$\displaystyle{\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^x=e}$
$\displaystyle{\lim_{x\to \infty}\frac{x^m}{e^x}=0,\quad \text{for any } m}$
$\displaystyle{\lim_{m\to \infty}\frac{(\ln x)^m}{m}=0}$
$\displaystyle{\lim_{x\to \infty}\frac{\ln )x+1)}{x}=1}$

Continuity of a Function

A single valued function, $f(x)$, is continuous at the point $a$ if and only if

  1. $f(a)$ exists
  2. $\displaystyle{\lim_{x\to a}f(x)}$ exists
  3. $\displaystyle{\lim_{x\to a}f(x)}=f(a)$

A single valued function, $f(x)$, is continuous on an interval $(a,b)$ or $[a,b]$ if and only if it is continuous at each point of the interval.

A single valued function, $f(x)$, has a discontinuity of the first kind at the point $x=a$ if the left hand and right hand limit of $f(x)$ at $a$ exist but are not equal. That is if $$\lim_{x\to a^-}f(x)\ne\lim_{x\to a^+}f(x).$$

A single valued function, $f(x)$, is piecewise-continuous on a given interval $(a,b)$ or $[a,b]$ if and only if $f(x)$ is continuous throughout this interval except for a finite number of discontinuities of the first kind.

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