Definition of Continuity - Continuity of a Function at a Point

If `f(x)=[tan(pi/4+x)]^(1/x), `

           = k                        ,for x=0

is continuous at x=0 , find k.

If f (x) = x 2 + a, for x ≥ 0 ` =2sqrt(x^2+1)+b, ` is continuous at x = 0, find a and b.

if the function

`f(x)=k+x, for x<1`

       `=4x+3, for x>=1`

id continuous at x=1 then k=

(a) 7

(b) 8

(c) 6

(d) -6

Examine continuity of the function f(x) at x = 0, where

`f(x) = (10^x + 7^x - 14^x - 5^x)/(1-cos 4x) , " for " x != 0`

`= 10/7 , " for"  x = 0`

Discuss the continuity of the following function, at x = 0.

`f(x)=x/|x|,for x ne0`

`=1,`for `x=0`

If `f(x) =(e^(x^2)-cosx)/x^2`, for x= 0, is continuous at x = 0, find f(0).

If the function `f(x)=(5^sinx-1)^2/(xlog(1+2x))`  for x ≠ 0 is continuous at x = 0, find f (0).

if  `f(x) = (x^2-9)/(x-3) + alpha`               for x> 3

           =5,                                     for x = 3

          `=2x^2+3x+beta`,             for x < 3

is continuous at x  = 3, find α and β.