Balbharati solutions for Mathematics and Statistics 1 (Commerce) [English] 11 Standard Maharashtra State Board chapter 8 - Continuity [Latest edition]

Examine the continuity of f(x) = x³ + 2x² − x − 2 at x = –2.

Examine the continuity of f(x) = `(x^2 - 9)/(x - 3)` on R.

Examine whether the function is continuous at the points indicated against them:
f(x) = x³ − 2x + 1,         for x ≤ 2
      = 3x − 2,                 for x > 2, at x = 2

Examine whether the function is continuous at the points indicated against them:
f(x) = `(x^2 + 18x - 19)/(x - 1)`        for x ≠ 1

      = 20                               for x = 1, at x = 1

Test the continuity of the following function at the points indicated against them:

`f(x) = (sqrt(x - 1) - (x - 1)^(1/3))/(x - 2)`  for x ≠ 2

         = `1/5`                                  for x = 2, at x = 2

Test the continuity of the following function at the points indicated against them:

`"f"(x) = (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))`  for x ≠ 2
         = – 24                               for x = 2, at x = 2

Test the continuity of the following function at the points indicated against them:

f(x) = 4x + 1,              for x ≤ 3

      = `(59 - 9x)/3`,        for x > 3 at x = `8/3`.

Test the continuity of the following function at the points indicated against them:

f(x) = `(x^3 - 27)/(x^2 - 9)`  for 0 ≤ x <3

      = `9/2`             for 3 ≤ x ≤ 6, at x = 3

If `f(x) = (24^x - 8^x - 3^x + 1)/(12^x - 4^x - 3^x + 1)` for x ≠ 0

            = k,                               for x = 0
is continuous at x = 0, find k.

If `f(x) = (5^x + 5^-x - 2)/(x^2)`  for x ≠ 0

          = k                            for x = 0
is continuous at x = 0, find k

For what values of a and b is the function

f(x) = ax + 2b + 18    for x ≤ 0

= x² + 3a − b            for 0 < x ≤ 2

= 8x – 2                     for x > 2,

continuous for every x ?

For what values of a and b is the function

`f(x) = (x^2 - 4)/(x - 2)`      for x < 2

= ax² − bx + 3         for 2 ≤ x < 3
= 2x – a + b             for x ≥ 3
continuous in its domain.

Discuss the continuity of the following function at the point(s) or in the interval indicated against them:

f(x) = 2x² − 2x + 5  for 0 ≤ x < 2

= `(1 - 3x - x^2)/(1 - x)`     for 2 ≤ x < 4

= `(7 - x^2)/(x - 5)`  for 4 ≤ x ≤ 7 on its domain.

Discuss the continuity of the following function at the point(s) or in the interval indicated against them.

`f(x) = (3^x + 3^-x - 2)/x^2`  for x ≠ 0.

= (log3)²                         for x = 0 at x = 0

Discuss the continuity of the following function at the point(s) or in the interval indicated against them:

`f(x) = (5^x - e^x)/(2x)`  for x ≠ 0

= `1/2`(log5−1)         for x = 0 at x = 0

f(x) = `(sqrt(x + 3) - 2)/(x^3 - 1)`  for x ≠ 1

= 2    for x = 1, at x = 1.

`f(x) = (log x - log 3)/(x - 3)` for x ≠ 3

= 3                                for x = 3, at x = 3.

Find k if the following function is continuous at the points indicated against them:

`f(x) = ((5x - 8)/(8 - 3x))^(3/(2x - 4)`  for x ≠ 2
         = k                           for x = 2 at x = 2.

Find k if the following function is continuous at the points indicated against them:

`f(x) = (45^x - 9^x - 5^x + 1)/((k^x - 1)(3^x - 1))`  for x ≠ 0

         = `2/3`                  for x = 0, at x = 0

Find k if the following function is continuous at the points indicated against them:

`f(x) = (1 + kx)^(1/x)`  , for x ≠ 0

       = `e^(3/2)`                  ,  for x = 0, at x = 0

Find a and b if the following function is continuous at the point indicated against them.

`f(x) = x^2 + a`    , for x ≥ 0

= `2sqrt(x^2 + 1) + b` , for x < 0 and
f(1) = 2 is continuous at x = 0

Find a and b if the following function is continuous at the point indicated against them.

`f(x) = (x^2 - 9)/(x - 3) + "a"`         , for x > 3
        = 5                             , x = 3
        = 2x² + 3x + b          , for x < 3
is continuous at x = 3

Find a and b if the following function is continuous at the point indicated against them.

`f(x) = (32^x - 1)/(8^x - 1) + a` , for x > 0
         = 2                      , for x = 0
         = x + 5 − 2b       , for x < 0
is continuous at x = 0