Continuity in the Domain of the Function

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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

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 ?

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

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

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

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.

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.

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.