Question

If function f(x) = `{{:((asinx + btanx - 3x)/x^3,",", x ≠ 0),(0,",", x = 0):}` is continuous at x = 0 then (a² + b²) is equal to ______.

Options

• 4.00

• 5.00

• 6.00

• 7.00

Answer 1

If function f(x) = `{{:((asinx + btanx - 3x)/x^3,",", x ≠ 0),(0,",", x = 0):}` is continuous at x = 0 then (a² + b²) is equal to 5.00.

Explanation:

f(x) = `{{:((asinx + btanx - 3x)/x^3,",", x ≠ 0),(0,",", x = 0):}`

`lim_(x→0) (a(x - x^3/(3!) + x^5/(5!)) + b(x + x^3/3 + (2x^5)/15) - 3x)/x^3`

Limit exist if a + b – 3 = 0  ......(i)

and `lim_(x→0) f(x)` = `f(0)` = 0

`-a/6 + b/3` = 0  ......(ii)

From (i) and (ii)

a = 2, b = 1

∴ a² + b² = 5.00