# 12 - Limit Continuity and Differentiability Questions Answers

Joshi sir comment
Joshi sir comment

Find

lim nāā  āæā(n!) / n

Joshi sir comment

Sir , i have solved   limx-->0    (sin-1x  - tan-1x )/x. plz have a look on others.

Joshi sir comment

limx-->0    (sin-1x  - tan-1x )/x3

use D L Hospital rule or expansion of sin-1x and tan-1x to solve it

no of solutions of  e-x^2/2 - x2 = 0 are

Joshi sir comment

two intersecting points means two solutions

lim x»infinity( ((x+1)(x+2)(x+3)(x+4))^¼ - x )

Joshi sir comment

multiply with the conjugate in numerator again and again you will get

limx->((x+1)(x+2)(x+3)(x+4) - x4)/((x+1)(x+2)(x+3)(x+4))1/4 + x )*((x+1)(x+2)(x+3)(x+4))1/2 + x2 )

after solving the numerator get a 3 power expression of x then take 3 power of x common, similarly take x and 2 power of x from the first and second denominator terms

lim n--> ∞  4^n/n!

Joshi sir comment

limn->∞ 4n/ n!

= limn->∞ [4/1][4/2][4/3][4/4][4/5][4/6][4/7]................................[4/n]

besides first four bracket, all brackets are real numbers less than 1 so their product will be zero for n tends to infinite

Importance High!!!!!!!!!!!!!!!!!

Myself Rajeev Shrivastava, I am putting a signs series in front of you which based on numbers from 01 to 100. Actually I want to know that what reason of behind Approx equability of signs is

(++), (+-), (--) & (-+) at end of the month or year between the below mentioned.

Let's define a kind of mapping:
0↔1
2↔3
4↔5
6↔7
8↔9

(+) means: an Even number
(-) means: an Odd number

I simply say: for any EO (+-) it exists an OE (-+) which is paired to.
And that's true since each E is mapped to a O and vice versa (1 to 1 relation where "f(f(x))=x" ) !
E.g.: 29 ↔ 83

There is 25 numbers in the EO (+-) group, so 25 in OE (+-).
There is 25 numbers in the EE (++) group, so 25 in OO (--).

Picking a number at random between 01 and 100 inclusive is choosing equitably in EO, OE, EE or OO group. (25% for each)

So, I really need of your help to solve a puzzle please. I just am trying to make you understand what type of solution I need:

There are some results for your review:

 MONTH DATE DAY FIRST_SERIES SECOND_SERIES FIRST_SERIES SECOND_SERIES APR 1 SUN + - - + +- -+ APR 2 MON + + - + ++ -+ APR 3 TUE + - + - +- +- APR 4 WED + + + + ++ ++ APR 5 THU - + - + -+ -+ APR 6 FRI + + + - ++ +- APR 7 SAT + - - + +- -+ APR 8 SUN - + + - -+ +- APR 9 MON - + - + -+ -+ APR 10 TUE + + + + ++ ++ APR 11 WED + + - - ++ -- APR 12 THU - + - - -+ -- APR 13 FRI + + + + ++ ++ APR 14 SAT + - + - +- +- APR 15 SUN + - + - +- +- APR 16 MON - - + + -- ++ APR 17 TUE - - + + -- ++ APR 18 WED APR 19 THU APR 20 FRI APR 21 SAT APR 22 APR 23

(Complete sheets for the year 2011 and 2012 are attached with this mail)

You can see that there are two times falls in a day of pair make by even (+) and odd (-) signs, i.e. (++), (+-), (--) & (-+) as above said.

The total of pair combination signs become approx equal to each other at the end of the month or year (every year- so, I am sending you attachment to go through the year’s result).

By this analysis:

1-     I want to know that what is the relation between current falling signs and past fell signs.

2-     How can I come to know, what type of combination of sign would be any particular date or day.

3-     As you can seen on 22nd and 23rd April’2012 I don’t know what combination of sign is.

4-     There is surety that there is some relation between current First fall of combination of signs and past fall or First fall of combination of signs and Second fall of combination of signs.

Please help me to how come to know what type of fall may be for next day by reviewing past or first combination of signs. So reply or elaborate me with an example.

I would be very grateful to you till entire life.

Regards

RAJEEV SHRIVASTAVA

Importance High!!!!!!!!!!!!!!!!!

Myself Rajeev Shrivastava, I am putting a signs series in front of you which based on numbers from 01 to 100. Actually I want to know that what reason of behind Approx equability of signs is

(++), (+-), (--) & (-+) at end of the month or year between the below mentioned.

Let's define a kind of mapping:
0↔1
2↔3
4↔5
6↔7
8↔9

(+) means: an Even number
(-) means: an Odd number

I simply say: for any EO (+-) it exists an OE (-+) which is paired to.
And that's true since each E is mapped to a O and vice versa (1 to 1 relation where "f(f(x))=x" ) !
E.g.: 29 ↔ 83

There is 25 numbers in the EO (+-) group, so 25 in OE (+-).
There is 25 numbers in the EE (++) group, so 25 in OO (--).

Picking a number at random between 01 and 100 inclusive is choosing equitably in EO, OE, EE or OO group. (25% for each)

So, I really need of your help to solve a puzzle please. I just am trying to make you understand what type of solution I need:

There are some results for your review:

 MONTH DATE DAY FIRST_SERIES SECOND_SERIES FIRST_SERIES SECOND_SERIES APR 1 SUN + - - + +- -+ APR 2 MON + + - + ++ -+ APR 3 TUE + - + - +- +- APR 4 WED + + + + ++ ++ APR 5 THU - + - + -+ -+ APR 6 FRI + + + - ++ +- APR 7 SAT + - - + +- -+ APR 8 SUN - + + - -+ +- APR 9 MON - + - + -+ -+ APR 10 TUE + + + + ++ ++ APR 11 WED + + - - ++ -- APR 12 THU - + - - -+ -- APR 13 FRI + + + + ++ ++ APR 14 SAT + - + - +- +- APR 15 SUN + - + - +- +- APR 16 MON - - + + -- ++ APR 17 TUE - - + + -- ++ APR 18 WED APR 19 THU APR 20 FRI APR 21 SAT APR 22 APR 23

(Complete sheets for the year 2011 and 2012 are attached with this mail)

You can see that there are two times falls in a day of pair make by even (+) and odd (-) signs, i.e. (++), (+-), (--) & (-+) as above said.

The total of pair combination signs become approx equal to each other at the end of the month or year (every year- so, I am sending you attachment to go through the year’s result).

By this analysis:

1-     I want to know that what is the relation between current falling signs and past fell signs.

2-     How can I come to know, what type of combination of sign would be any particular date or day.

3-     As you can seen on 22nd and 23rd April’2012 I don’t know what combination of sign is.

4-     There is surety that there is some relation between current First fall of combination of signs and past fall or First fall of combination of signs and Second fall of combination of signs.

Please help me to how come to know what type of fall may be for next day by reviewing past or first combination of signs. So reply or elaborate me with an example.

I would be very grateful to you till entire life.

Regards

RAJEEV SHRIVASTAVA

Joshi sir comment

If you are making the sequence on the basis of dates then you should take signs accordingly then you can make this sequence for many years

for example if you take 23 june then first sign series will be +-, + for even number and - for odd number , now as conversion rule as you gave in your explanation 23 will be converted to 32 so second series will be -+

similarly you can make it for any date

for example 29 december first series will be +- and its conversion is 38 so next series will be -+

if you are thinking something else then explain your question correctly

ABC is an isosceles triangle inscribed in a circle of radius r. If  AB=AC and h is the altitude from A to BC. and P and φ denote the perimeter and area of the triangle respectively, then  lim n−» 0   φ/p³ is equal to??

Joshi sir comment

first draw a cirle and triange inside it , consider angle B and angle C asα so angle A = 180-2α, let centre of the circle is O and D is the point in the line BC where altitude meets in the line BC, given that altitude = h and radius = r

so AB = AC = h cosecα and BC = 2 h cotα so

p = 2 h (cosecα + cotα)

and φ = 1/2  2 h cotα h = h2 cotαa

and in triangle OBD angle OBD = 2α - 90    so cos(2α - 90) = h cotα /r   implies that h = 2 r sin2α

now i think limit will be based on h not n

lim h−» 0   φ/p³  = 1/128r

for getting solution put p and φ first then put h in terms of r, you will get an equation based on α and r,

convert limh->0  to  limα->0     because when h will be 0, α will also be 0

limit  n tends to ∞

then

(x^n)/(n!) equals