HSC Arts कक्षा ११ - Tamil Nadu Board of Secondary Education Question Bank Solutions

Evaluate the following limits:

`lim_(x  -> oo) 3/(x - 2) - (2x + 11)/(x^2 + x - 6)`

Evaluate the following limits:

`lim_(x -> oo) (x^3 + x)/(x^4 - 3x^2 + 1)`

Evaluate the following limits:

`lim_(x -> oo) (x^4 - 5x)/(x^2 - 3x + 1)`

Evaluate the following limits:

`lim_(x -> oo) (1 + x - 3x^3)/(1 + x^2 +3x^3)`

Evaluate the following limits:

`lim_(x ->oo) (x^3/(2x^2 - 1) - x^2/(2x + 1))`

Show that `lim_("n" -> oo) (1 + 2 + 3 + ... + "n")/(3"n"^2 + 7n" + 2) = 1/6`

Show that  `lim_("n" -> oo) (1^2 + 2^2 + ... + (3"n")^2)/((1 + 2 + ... + 5"n")(2"n" + 3)) = 9/25`

Show that `lim_("n" -> oo) 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/("n"("n" + 1))` = 1

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"/((alpha"S" + beta)` where `alpha` and `beta` are positive constants. Show that this function predicts approximately constant recruitment when the number of spawners is sufficiently large

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) = `(30"t")/(200 + "t")`. What happens to the concentration as t → ∞?

Evaluate the following limits:

`lim_(x -> oo)(1 + 1/x)^(7x)`

Evaluate the following limits:

`lim_(x -> 0)(1 + x)^(1/(3x))`

Evaluate the following limits:

`lim_(x -> oo)(1 + "k"/x)^("m"/x)`

Evaluate the following limits:

`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`

Evaluate the following limits:

`lim_(x -> oo) (1 + 3/x)^(x + 2)`

Evaluate the following limits:

`lim_(x -> 0) (sin^3(x/2))/x^2`

Evaluate the following limits:

`lim_(x -> 0) (sinalphax)/(sinbetax)`

Evaluate the following limits:

`lim_(x -> 0) (tan 2x)/(sin 5x)`

Evaluate the following limits:

`lim_(alpha -> 0) (sin(alpha^"n"))/(sin alpha)^"m"`

Evaluate the following limits:

`lim_(x -> 0) (sin("a" + x) - sin("a" - x))/x`