# LIMITS AT INFINITY WITH VARIABLE N

## About "Limits At Infinity With Variable n"

Limits At Infinity With Variable n :

Here we are going to see how to evaluate limits at infinity with variable n.

## Limits At Infinity With Variable n - Examples

Question 1 :

Show that lim n-> (1 + 2 + 3 + .........+ n) / (3n2+ 7n +2)  =  1/6

Solution :

=  lim n-> (1 + 2 + 3 + .........+ n) / (3n2+ 7n +2)

By using the formula for n natural numbers.

=  lim n-> (n(n+1)/2) / (3n2+ 7n + 2)

=  lim n-> (n2 + n) / 2(3n2+ 7n + 2)

=  lim n-> (1 + 1/n) / (6 + 14/n + 2/n2

By applying the limit, we get

=  1/6

Hence the value of Show that lim n-> (1 + 2 + 3 + .........+ n) / (3n2+ 7n +2) is 1/6.

Question 2 :

Show that lim n-> (12+22+.........+(3n)2) / (1+2+....+5n)(2n+3)  =  9/25

Solution :

=  lim n-> (12+22+.........+(3n)2) / (1+2+....+5n)(2n+3)

=  lim n->∞ (3n(3n+1)(3n+2)/6) / (5n(5n+1)/2)(2n+3)

=  lim n->∞ (3n(3n+1)(3n+2)/6) / (5n(5n+1)(2n+3)/2)

=  lim n->∞ 3n(2)(3n+1)(3n+2)/(6(5n)(5n+1)(2n+3)

=  lim n->∞ (3n+1)(3n+2) / 5(5n+1)(2n+3)

=  lim n->∞ (1/5)(9n2+9n+2) / (10n2+17n+3)

Let us divide numerator and denominator by n2

=  lim n->(1/5)(9 + 9/n + 2/n2) / (10 + 5/n + 3/n2)

By applying the limit, we get

=  (1/5) (9/10)

=  9/50

Hence the value of lim n-> (12+22+.........+(3n)2) / (1+2+....+5n)(2n+3) is 9/50.

Question 3 :

Show that lim n-> (1/1⋅2 + 1/2⋅3 + 1/3⋅4+....+1/n(n+1))  = 1

Solution :

Let us write

1/1⋅2 as (1/1) - (1/2)  =  (2 - 1)/1⋅2  =  1/1⋅2 ----(1)

1/2⋅3  =  (1/2) - (1/3)  =  (3 - 2)/2⋅3  =  1/2⋅3 ----(2)

1/3⋅4  =  (1/3) - (1/4)  =  (4 - 3)/3⋅4  =  1/3⋅4 ----(3)

(1) + (2) + (3) ==>

=  (1/1)-(1/2)+(1/2)-(1/3)+(1/3)-(1/4)+........(1/n)- (1/(n+1))

=  1 - (1/(n+1))

=  lim n->∞  [1 - (1/(n+1))]

By applying the limit, we get

=  1

Hence the value of  lim n-> (1/1⋅2 + 1/2⋅3 + 1/3⋅4+....+1/n(n+1)) is 1.

Question 4 :

An important problem in fishery science is to estimate the number of fish presently spawning in streams and use this information to predict the number of mature fish or “recruits” that will return to the rivers during the reproductive period. If S is the number of spawners and R the number of recruits, “Beverton-Holt spawner recruit function” is R(S) = S/(αS + where a and b are positive constants. Show that this function predicts approximately constant recruitment when the number of spawners is sufficiently large.

Solution :

When the number of spawners is sufficiently large means S -∞

R(S) = S/(α S +

=  lim S->∞ S/(α S +

Dividing the equation by S, we get

=  lim S->1/(α + /S

By applying the limit, we get

1/α

Question 5 :

A tank contains 5000 litres of pure water. Brine (very salty water) that contains 30 grams of salt per litre of water is pumped into the tank at a rate of 25 litres per minute. The concentration of salt water after t minutes (in grams per litre) is C(t) = 30t/(200 + t).What happens to the concentration as  -∞

Solution :

To find the quantity of concentration as  -∞

C(t) = lim x->∞ 30t/(200 + t)

C(t) = lim x->∞ 30/(200/t + 1)

=  30/1

=  30 After having gone through the stuff given above, we hope that the students would have understood, "Limits At Infinity With Variable n"

Apart from the stuff given in "Limits At Infinity With Variable n" if you need any other stuff in math, please use our google custom search here.

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 