Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 11 TN Board chapter 9 - Differential Calculus - Limits and Continuity [Latest edition]

In problems 1 – 6, using the table estimate the value of the limit.

`lim_(x -> 2) (x - 2)/(x^2 - x - 2)`

In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 2) (x - 2)/(x^2 - 4)`

In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) (sqrt(x + 3) - sqrt(3))/x`

In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> - 3) (sqrt(1 - x) - 2)/(x + 3)`

In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) sin x/x`

In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) (cos x - 1)/x`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 3) (4 - x)`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 1) (x^2 + 2)`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 2) f(x)` where `f(x) = {{:(4 - x",", x ≠ 2),(0",", x = 2):}`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 1) f(x)` where `f(x) = {{:(x^2 + 2",", x ≠ 1),(1",", x = 1):}`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 3) 1/(x - 3)`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 5) |x - 5|/(x - 5)`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 1) sin pi x`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 0) sec x`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> x/2) tan x`

Sketch the graph of f, then identify the values of x₀ for which `lim_(x -> x_0)` f(x) exists.

f(x) = `{{:(x^2",", x ≤ 2),(8 - 2x",", 2 < x < 4),(4",", x ≥ 4):}`

Sketch the graph of f, then identify the values of x₀ for which `lim_(x -> x_0)` f(x) exists.

f(x) = `{{:(sin x",", x < 0),(1 - cos x",", 0 ≤ x ≤ pi),(cos x",", x > pi):}`

Sketch the graph of a function f that satisfies the given value:

f(0) is undefined

`lim_(x -> 0) f(x)` = 4

f(2) = 6

`lim_(x -> 2) f(x)` = 3

Sketch the graph of a function f that satisfies the given value:

f(– 2) = 0

f(2) = 0

`lim_(x -> 2) f(x)` = 0

`lim_(x -> 2) f(x)` does not exist.

Write a brief description of the meaning of the notation `lim_(x -> 8) f(x)` = 25

If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2?

If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning

Evaluate : `lim_(x -> 3) (x^2 - 9)/(x - 3)` if it exists by finding `f(3^-)` and `f(3^+)`

Verify the existence of `lim_(x -> 1) f(x)`, where `f(x) = {{:((|x - 1|)/(x - 1)",",  "for"  x ≠ 1),(0",",  "for"  x = 1):}`

Evaluate the following limits:

`lim_(x -> 2) (x^4 - 16)/(x - 2)`

Evaluate the following limits:

`lim_(x ->) (x^"m" - 1)/(x^"n" - 1)`, m and n are integers

Evaluate the following limits:

`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)`

Evaluate the following limits:

`lim_("h" -> 0) (sqrt(x + "h") - sqrt(x))/"h", x > 0`

Evaluate the following limits:

`lim_(x -> 5) (sqrt(x + 4) - 3)/(x - 5)`

Evaluate the following limits:

`lim_(x -> 2) (1/x - 1/2)/(x - 2)`

Evaluate the following limits:

`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x))`

Evaluate the following limits:

`lim_(x -> 0) (sqrt(x^2 + 1) - 1)/(sqrt(x^2 + 16) - 4)`

Evaluate the following limits:

`lim_(x -> 0) (sqrt(1 + x) - 1)/x`

Evaluate the following limits:

`lim_(x -> 1) (root(3)(7 + x^3) - sqrt(3 + x^2))/(x - 1)`

Evaluate the following limits:

`lim_(x -> 2) (2 - sqrt(x + 2))/(root(3)(2) - root(3)(4 - x))`

Evaluate the following limits:

`lim_(x - 0) (sqrt(1 + x^2) - 1)/x`

Evaluate the following limits:

`lim_(x -> 0) (sqrt(1 - x) - 1)/x^2`

Evaluate the following limits:

`lim_(x -> 5) (sqrt(x - 1) - 2)/(x - 5)`

Evaluate the following limits:

`lim_(x -> "a") (sqrt(x - "b") - sqrt("a" - "b"))/(x^2 - "a"^2) ("a" > "b")`

Find the left and right limits of f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2

Find the left and right limits of f(x) = tan x at x = `pi/2`

Evaluate the following limits:

`lim_(x -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9))`

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`

Evaluate the following limits:

`lim_(x -> 0) (sqrt(x^2 + "a"^2) - "a")/(sqrt(x^2 + "b"^2) - "b")`

Evaluate the following limits:

`lim_(x -> 0) (2 "arc"sinx)/(3x)`

Evaluate the following limits:

`lim_(x-> 0) (1 - cos x)/x^2`

Evaluate the following limits:

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

Evaluate the following limits:

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

Evaluate the following limits:

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

Evaluate the following limits:

`lim_(x -> 0) (1 - cos^2x)/(x sin2x)`

Evaluate the following limits:

`lim_(x -> oo) x [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)]`

Evaluate the following limits:

`lim_(x - oo){x[log(x + "a") - log(x)]}`

Evaluate the following limits:

`lim_(x -> pi) (sin3x)/(sin2x)`

Evaluate the following limits:

`lim_(x -> pi) (1 + sinx)^(2"cosec"x)`

Evaluate the following limits:

`lim_(x -> 0) (sqrt(2) - sqrt(1 + cosx))/(sin^2x)`

Evaluate the following limits:

`lim_(x -> 0) (sqrt(1 + sinx) - sqrt(1 - sinx))/tanx`

Evaluate the following limits:

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

Evaluate the following limits:

`lim_(x -> 0) ("e"^x - "e"^(-x))/sinx`

Evaluate the following limits:

`lim_(x -> 0) ("e"^("a"x) - "e"^("b"x))/x`

Evaluate the following limits:

`lim_(x -> ) (sinx(1 - cosx))/x^3`

Evaluate the following limits:

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

Prove that f(x) = 2x² + 3x - 5 is continuous at all points in R

Examine the continuity of the following:

x + sin x

Examine the continuity of the following:

x² cos x

Examine the continuity of the following:

e^x tan x

Examine the continuity of the following:

e^2x + x²

Examine the continuity of the following:

x . log x

Examine the continuity of the following:

`sinx/x^2`

Examine the continuity of the following:

`(x^2 - 16)/(x + 4)`

Examine the continuity of the following:

|x + 2| + |x – 1|

Examine the continuity of the following:

`|x - 2|/|x + 1|`

Examine the continuity of the following:

cot x + tan x

Find the points of discontinuity of the function f, where `f(x) = {{:(4x + 5",",  "if",  x ≤ 3),(4x - 5",",  "if",  x > 3):}`

Find the points of discontinuity of the function f, where `f(x) = {{:(x + 2",",  "if",  x ≥ 2),(x^2",",  "if",  x < 2):}`

Find the points of discontinuity of the function f, where `f(x) = {{:(x^3 - 3",",  "if"  x ≤ 2),(x^2 + 1",",  "if"  x < 2):}`

Find the points of discontinuity of the function f, where `f(x) = {{:(sinx",",  0 ≤ x ≤ pi/4),(cos x",", pi/4 < x < pi/2):}`

At the given point x₀ discover whether the given function is continuous or discontinuous citing the reasons for your answer:

x₀ = 1, `f(x) = {{:((x^2 - 1)/(x - 1)",", x ≠ 1),(2",", x = 1):}`

At the given point x₀ discover whether the given function is continuous or discontinuous citing the reasons for your answer:

x₀ = 3, `f(x) = {{:((x^2 - 9)/(x - 3)",", "if"  x ≠ 3),(5",", "if"  x = 3):}`

Show that the function `{{:((x^3 - 1)/(x - 1)",",  "if"  x ≠ 1),(3",",  "if"  x = 1):}` is continuous om `(- oo, oo)`

For what value of `alpha` is this function `f(x) = {{:((x^4 - 1)/(x - 1)",",  "if"  x ≠ 1),(alpha",",  "if"  x = 1):}` continuous at x = 1?

Let `f(x) = {{:(0",",  "if"  x < 0),(x^2",",  "if"  0 ≤ x ≤ 2),(4",",  "if"  x ≥ 2):}`. Graph the function. Show that f(x) continuous on `(- oo, oo)`

If f and g are continuous functions with f(3) = 5 and `lim_(x -> 3) [2f(x) - g(x)]` = 4, find g(3)

Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f.

`f(x) = {{:(2x + 1",",  "if"  x ≤ - 1),(3x",",  "if"  - 1 < x < 1),(2x - 1",",  "if"  x ≥ 1):}`

Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f.

`f(x) = {{:((x - 1)^3",",  "if"  x < 0),((x + 1)^3",",  "if"  x ≥ 0):}`

A function f is defined as follows:

`f(x) = {{:(0,  "for"  x < 0;),(x,  "for"  0 ≤ x ≤ 1;),(- x^2 +4x - 2, "for"  1 ≤ x ≤ 3;),(4 - x,  "for"  x ≥ 3):}`
Is the function continuous?

Which of the following functions f has a removable discontinuity at x = x₀? If the discontinuity is removable, find a function g that agrees with f for x ≠ x₀ and is continuous on R.

`f(x) = (x^2 - 2x - 8)/(x + 2), x_0` = – 2

Which of the following functions f has a removable discontinuity at x = x₀? If the discontinuity is removable, find a function g that agrees with f for x ≠ x₀ and is continuous on R.

`f(x) = (x^3 + 64)/(x + 4), x_0` = – 4

Which of the following functions f has a removable discontinuity at x = x₀? If the discontinuity is removable, find a function g that agrees with f for x ≠ x₀ and is continuous on R.

`f(x) = (3 - sqrt(x))/(9 - x), x_0` = 9

Find the constant b that makes g continuous on `(- oo, oo)`.

`g(x) = {{:(x^2 - "b"^2,"if"  x < 4),("b"x + 20,  "if"  x ≥ 4):}`

Consider the function  `f(x) = x sin  pi/x`. What value must we give f(0) in order to make the function continuous everywhere?

The function `f(x) = (x^2 - 1)/(x^3 - 1)` is not defined at x = 1. What value must we give f(1) inorder to make f(x) continuous at x =1?

State how continuity is destroyed at x = x₀ for the following graphs.

State how continuity is destroyed at x = x₀ for the following graphs.

State how continuity is destroyed at x = x₀ for the following graphs.

State how continuity is destroyed at x = x₀ for the following graphs.

Choose the correct alternative:

`lim_(x -> oo) sinx/x`

Choose the correct alternative:

`lim_(x - pi/2) (2x - pi)/cos x`

Choose the correct alternative:

`lim_(x -> 0) sqrt(1 - cos 2x)/x`

Choose the correct alternative:

`lim_(theta -> 0) (sinsqrt(theta))/(sqrt(sin theta)`

Choose the correct alternative:

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

Choose the correct alternative:

`lim_(x - oo) sqrt(x^2 - 1)/(2x + 1)` =

Choose the correct alternative:

`lim_(x -> 0) ("a"^x - "b"^x)/x` =

Choose the correct alternative:

`lim_(x -> 0) (8^x - 4x - 2^x + 1^x)/x^2` =

Choose the correct alternative:

If `f(x) = x(- 1)^([1/x])`, x ≤ 0, then the value of `lim_(x -> 0) f(x)` is equal to

Choose the correct alternative:

`lim_(x -> 3) [x]` =

Choose the correct alternative:

Let the function f be defined by `f(x) = {{:(3x, 0 ≤ x ≤ 1),(-3 + 5, 1 < x ≤ 2):}`, then

Choose the correct alternative:

If f : R → R is defined by `f(x) = [x - 3] + |x - 4|` for x ∈ R then `lim_(x -> 3^-) f(x)` is equal to

Choose the correct alternative:

`lim_(x -> 0) (x"e"^x - sin x)/x` is

Choose the correct alternative:

If `lim_(x -> 0) (sin "p"x)/(tan 3x)` = 4, then the value of p is

Choose the correct alternative:

`lim_(alpha - pi/4) (sin alpha - cos alpha)/(alpha - pi/4)` is

Choose the correct alternative:

`lim_(x -> oo) (1/"n"^2 + 2/"n"^2 + 3/"n"^2 + ... + "n"/"n"^2)` is

Choose the correct alternative:

`lim_(x -> 0) ("e"^(sin x) - 1)/x` =

Choose the correct alternative:

`lim_(x -> 0) ("e"^tanx - "e"^x)/(tan x - x)` =

Choose the correct alternative:

The value of `lim_(x -> 0) sinx/sqrt(x^2)` is

Choose the correct alternative:

The value of `lim_(x -> "k") x - [x]`, where k is an integer is

Choose the correct alternative:

At x = `3/2` the function f(x) = `|2x - 3|/(2x - 3)` is

Choose the correct alternative:

Let f : R → R be defined by `f(x) = {{:(x, x  "is irrational"),(1 - x, x  "is rational"):}` then f is

Choose the correct alternative:

The function `f(x) = {{:((x^2 - 1)/(x^3 + 1), x ≠ - 1),("P", x = -1):}` is not defined for x = −1. The value of f(−1) so that the function extended by this value is continuous is

Choose the correct alternative:

Let f be a continuous function on [2, 5]. If f takes only rational values for all x and f(3) = 12, then f(4.5) is equal to

Choose the correct alternative:

Let a function f be defined by `f(x) = (x - |x|)/x` for x ≠ 0 and f(0) = 2. Then f is