LIMIT: IMPORTANT DEFINITIONS AND RESULTS - I

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IIT-JEE 2009

LIMIT: IMPORTANT DEFINITIONS AND RESULTS - I

Author: anuraagchandra

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IMPORTANT DEFINITIONS AND RESULTS

1. Meaning of 'tends to':

$x \rightarrow 2-0$ or $x \rightarrow 2-$ . This is read as x tends to 2 from left. This means x is very close to 2 but it is always less than 2.
$x \rightarrow 2+0$ or $x \rightarrow 2+$ . This means x is very close to 2 but it is always greater than 2.
$x \rightarrow 2.$ This is read as x tends to 2. This means x is very close to 2 but it is never equal to 2.

2. Concept of infinity: $x=\infty$ or $x \rightarrow \infty$ means:

x is larger than any number however large.
x is not a fixed number.

3. Indeterminate or meaningless forms: Following seven forms are indeterminate

$\frac {0}{0}$ Form: Here numerator $\rightarrow 0$ and denominator $\rightarrow 0$ .
$\frac {\infty} {\infty}$ Form: Here num. $\rightarrow \infty$ and denom. $\rightarrow \infty$
$0, \infty form$ Here one factor tends to zero and other factor tends to $\infty$
$\infty- \infty form$
$0 ^{\circ}$ $0^{\circ}$ form: Here base $\rightarrow 0$ and power $\rightarrow 0$
$\infty ^ {\circ}$ form: Here base $\rightarrow \infty$ and power tends to 0.
$1 ^ {\infty}form:$ Here base $\rightarrow 1$ and power tends to $\infty$

4. Limit of function:A number l is said to be the limit of a function f (x) as x tends to a if as x approaches a, f (x) approaches l or f (x) is equal to l. This number l should be unique. In this case we write $\overset {}{\underset {x \rightarrow a}{Lt}} f(x)=l$ . Thus $\overset {}{\underset {x \rightarrow a}{Lt}} f(x)=l$ means as x approaches a, f (x) approaches l or f (x) = l.

Ex.

$\overset {}{\underset {x \rightarrow 2}{Lt}} (x+2) = 4$
$[\therefore as \quad x \rightarrow 2, x+2 \rightarrow 4]$
If f(x) = 2, then $\overset {}{\underset {x \rightarrow a}{Lt}} f(x)=2$

$[\therefore as \quad x \rightarrow a, f(x)=2]$
$\overset {}{\underset {x\rightarrow 2}{Lt}}\frac {x^2-4}{x-2}$ [Here indeterminate form is $\frac {0} {0}$ ]

$= \overset {}{\underset {x\rightarrow 2}{Lt}}\frac {(x-2)(x+2)}{x-2}$

$=\overset {}{\underset {x \rightarrow 2}{Lt}} (x+2) = 4 \qquad (\therefore as \quad x \rightarrow 2, x+2 \rightarrow 4)$

5. If one factor is zero and other tends to $\infty$

$0.\infty =0,$
But $0.\infty$ is indeterminate if one factor tends to 0 and other tends to $\infty$ .
$1 ^ {\infty} = 1$ , If base is equal to 1 and power tends to $\infty$ .
But $1 ^ {\infty}$ is indeterminate if base $\rightarrow 1$ and power tends to $\infty$ .
$\infty- \infty$ is indeterminate.
But $\overset {}{\underset {x \rightarrow \infty} {Lt}}(x ^ 2 - x ^ 2)= 0$
$\frac{\infty} {\infty}$ is indeterminate. But $\overset {}{\underset {x \rightarrow \infty}{Lt}}\frac{x ^ 2}{x ^ 2} = 1$

6. Properties of Infinity:

$\infty \pm C=\infty$
$\infty + \infty = \infty$
$\infty .\infty =\infty$
$\infty (- \infty)=- \infty , (- \infty)\infty =- \infty$
$\infty ^ {\infty}=\infty$
$C.\infty = \infty$ , C> $- \infty ,C<0=0 , C=0$
In fact $C.\infty \rightarrow \infty$ , if C > 0

$c.\infty \rightarrow -\infty$ , if C < 0 and $0.\infty = 0$
$C ^ {\infty} =\infty$ , if C > 1

= 0, if 0 â‰¤ C < 1

=1, if C = 1

In fact $c^{\infty} \rightarrow \infty$ if c > 1

$c ^{\infty} \rightarrow 0$ , if 0 < c < 1

$c ^{\infty} =0$ , if c = 0

$c ^{\infty} =1$ , if c = 1

- $\frac {0}{tends\quad to \quad zero}=0$
- $\frac {0}{0}$ is undefined
- $\frac {tends \quad to \quad zero}{0}$ is undefined
- $\frac {tends \quad to \quad zero}{tends \quad to \quad zero}$ is indeterminate
- $(tends \quad to \quad zero)^{\circ}=1$
- $0^{tends \quad to \quad 0 \quad form \quad right}=0$
- $0 ^{\circ}$ is undefined
- $(tends \quad to \quad zero )^{tends \quad to \quad zero}$ is indeterminate.
Whenever power is variable base is taken as positive, it can be negative only when power is a variable but an integer.
and
1. If , , then
  - $\overset {}{\underset{x \rightarrow a}{Lt}}\{ f(x)\pm g(x)\}$ $=\overset {}{\underset {x \rightarrow a}{Lt}} f(x)\pm$ $\overset {}{\underset {x \rightarrow a}{Lt}}$
  - $\overset {}{\underset {x\rightarrow a}{Lt}}\{ f(x)g(x)\}$ $=\overset {}{\underset {x\rightarrow a}{Lt}}f(x)$ $\overset {}{\underset {x\rightarrow a}{Lt}}g(x)$
  - $\overset {}{\underset {x\rightarrow a}{Lt}}\frac {f(x)}{g(x)}=$ $\frac {\overset{}{\underset{x\rightarrow a}{Lt}}f(x)}{\overset {}{\underset{x\rightarrow a}{Lt}}g(x)}$ $=\frac {l_1}{l_2}$ , where $l_2 \not =0$
  - If $P=\overset {}{\underset {x\rightarrow a}{Lt}}(f(x)) ^{g(x)}$ , then $log_e P=\overset {}{\underset {x\rightarrow a}{Lt}}g(x) \log \{f(x)\}$
2. If $\overset {}{\underset {x\rightarrow a}{Lt}}\{f(x)\pm g(x)\}$ exists, then it is not necessary that $\overset {}{\underset {x\rightarrow a}{Lt}} f.(x)$ and $\overset {}{\underset {x\rightarrow a}{Lt}}g(x)$ will exist.
3. If $\overset {}{\underset {x \rightarrow a}{Lt}}\{f(x)g(x)\}$ exists, then it is not necessary that $\overset {}{\underset {x \rightarrow a}{Lt}}f(x)$ and $\overset {}{\underset {x \rightarrow a}{Lt}}g(x)$ will exist.