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e`e^(-1)`1none of these

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BSolution :

Let `A underset(ntooo)lim ((n!)^(1//n))/(n)` then, <br> `log A=underset(ntooo)lim log((1*2*3..n)/(n^(n)))^(1//n)` <br> ` rArr log A=underset(ntooo)lim log((1)/(n)*(2)/(n)*(3)/(n)...(n)/(n))^(1//n)` <br> `rArrlog A=underset(ntooo)lim (1)/(n)underset(r=1)overset(n)sum log((r)/(n))` <br> `rArrlog A=underset(0)overset(1)intlog x dx= [ xlog x-x]_(0)^(1)=-1` <br> `:. A=e^(-1)`Transcript

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00:00 - 00:59 | the question is limit and approaches to infinity and fitted to the power 1 by N / M = this we need to find out there so let let a is equal to Limit and approaches to infinity and factorial to the power 1 by an / to solve this first we will simplify it like this we will write limit and approaches to infinity and factorial divided by and to the power and whole to the power 1 by Android nominate like this and and and and power 1 by and would get cancelled out so we can write like this now we will take Logon Woodside taking log on both sides taking log on both sides come |

01:00 - 01:59 | equal to Limit and infinity this becomes log of n factorial and can be written as well as one into 2 into 3 and products loan upto and and later we can write again and again this is and to the power and then multiplied and times can we eat like this and we have won by Power of Now lock has a property that if we have log m to the power and this is equal to an end to the above am ok power and cats X side like this right lobe of 10 is equal to Limit and approaches to infinity 1 by N |

02:00 - 02:59 | one my mm2 to buy an in 23 and 24 by Anshuman like this up to and by this week and simplify Android like this log of a is equal to Limit and approaches to infinity and times of also don't have a property that log of product of two numbers a and b is equal to log off in + log of this case we will use this year so this is product of these and number so we can simply write thesis submission from 1 to end of log of physical right now to solve this will change this limit |

03:00 - 03:59 | into integration show formula to do to do that is formula that will use here is an integer limit and approaches to infinity one by M summation Our Stars from x to Y O U R and B have limit of this from where we have numbers like and 1 by N and a string and we have emission from this to disturb this can be written as equal to integration into B of Accord ex BF share what we are doing VR VR replacing |

04:00 - 04:59 | omission by integration in place of RBI and we are replacing eggs are by an is replaced by X in place of 1 by an we are replacing the year 2001 when we are facing DF and their limits are a limit is equal to Limit and approaches to infinity the lower limit / and and b is equal to Limit and approaches to infinity the higher power limit of the this submission / n c using this formula we can write log of a is equal to log off in is equal to in place of 1 by and we will write TX and in place of |

05:00 - 05:59 | RBI and we will write text from this formula to this becomes log of x dx and also the commission part changes to intragel and lower limits are if you put hair use this formula so one by one and approaches Infinity this becomes here in this question we have a is 1 by an end and approaches to infinity Shoaib zero Limited integration lower limit is zero and similarly upper limit when we put bb5 is there an Subin will get equal to 1 then this is the simplified version of this question now so simply didn't integration of log x 10 x log x minus |

06:00 - 06:59 | x and put the values the limit start from zero to one so this becomes on putting one with this is one into log of 1 -1 and putting zero everything becomes zero so this is equal to log off one is also Hero this is equal to 1 from here log of a is equal to 1 from where we get value of day is it to the power minus 1 is equal to 1 by this is the value of the limit is the correct |

**Fundamental Theorem of Definite Integration**

**`int_a ^b f(x) dx = phi(b) - phi(a)`**

**Examples: `int_2 ^4 x / (x^2 + 1) dx`**

**Definite integration by substitution**

**Examples: `int_0 ^1 sin^-1 ((2x )/ (1 + x^2)) dx`**

**Property 1: Integration is independent of the change of variable. `int_a ^b f(x) dx = int_a ^b f(t) dt`**

**Property 2: If the limits of a definite integral are interchanged then its value changes. `int_a ^b f(x) dx = - int_b ^a f(x) dx`**

**Property 3: `int_a ^b f(x) dx = int_a ^c f(x)dx + int_c ^b f(x) dx`**

**Property 4: If `f(x)` is a continuous function on `[a,b]` then `int_a ^b f(x) dx = int_a ^b f(a+b-x) dx`**

**Property 5: If `f(x)` is a continuous function defined on `[0,a]` then `int_0 ^a f(x) dx = int_0^a f(a-x) dx`**