**Limits At Infinity With Variable n :**

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

**Question 1 :**

Show that lim _{n->}_{∞} (1 + 2 + 3 + .........+ n) / (3n^{2}+ 7n +2) = 1/6

**Solution :**

= lim _{n->}_{∞} (1 + 2 + 3 + .........+ n) / (3n^{2}+ 7n +2)

By using the formula for n natural numbers.

= lim _{n->}_{∞} (n(n+1)/2) / (3n^{2}+ 7n + 2)

= lim _{n->}_{∞} (n^{2} + n) / 2(3n^{2}+ 7n + 2)

= lim _{n->}_{∞} (1 + 1/n) / (6 + 14/n + 2/n^{2})

By applying the limit, we get

= 1/6

Hence the value of Show that lim _{n->}_{∞} (1 + 2 + 3 + .........+ n) / (3n^{2}+ 7n +2) is 1/6.

**Question 2 :**

Show that lim _{n->}_{∞} (1^{2}+2^{2}+.........+(3n)^{2}) / (1+2+....+5n)(2n+3) = 9/25

**Solution :**

= lim _{n->}_{∞} (1^{2}+2^{2}+.........+(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)(9n^{2}+9n+2) / (10n^{2}+17n+3)

Let us divide numerator and denominator by n^{2}

= lim _{n->}_{∞ }(1/5)(9 + 9/n + 2/n^{2}) / (10 + 5/n + 3/n^{2})

By applying the limit, we get

= (1/5) (9/10)

= 9/50

Hence the value of lim _{n->}_{∞} (1^{2}+2^{2}+.........+(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 + ᵦ) 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 + ᵦ)

= lim _{S->}_{∞ }S/(α S + ᵦ)

Dividing the equation by S, we get

= lim _{S->}_{∞ }1/(α + ᵦ/S)

By applying the limit, we get

= 1/α

Hence the answer is 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 t ** -> **∞

**Solution :**

To find the quantity of concentration as t ** -> **∞

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.

HTML Comment Box is loading comments...

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 **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**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**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**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 **

**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 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**