Question

If f(x) = `{{:((sin(p  +  1)x  +  sinx)/x,",", x < 0),(q,",", x = 0),((sqrt(x  +  x^2)  -  sqrt(x))/(x^(3//2)),",", x > 0):}`

is continuous at x = 0, then the ordered pair (p, q) is equal to ______.

Options

• `(-3/2, -1/2)`

• `(-1/2, 3/2)`

• `(-3/2, 1/2)`

• `(5/2, 1/2)`

Answer 1

If f(x) = `{{:((sin(p  +  1)x  +  sinx)/x,",", x < 0),(q,",", x = 0),((sqrt(x  +  x^2)  -  sqrt(x))/(x^(3//2)),",", x > 0):}`

is continuous at x = 0, then the ordered pair (p, q) is equal to `underlinebb((-3/2, 1/2)`.

Explanation:

f(x) = `{{:((sin(p  +  1)x  +  sinx)/x, x < 0),(q, x = 0  "is continuous at"  x = 0),((sqrt(x^2 +  x)  -  sqrt(x))/(x^(3/2)), x > 0) :}` 

Therefore, f(0^–) = f(0) = f(0)⁺  ...(i)

f(0^–) = `lim_(h rightarrow 0)f(0 - h)`

= `lim_(h rightarrow 0) (sin(p + 1)(-h) + sin(-h))/(-h)`

= `lim_(h rightarrow 0) [(-sin(p + 1)h)/(-h) + (sinh)/h]`

= = `lim_(h rightarrow 0) (sin(p + 1)h)/(h(p + 1)) xx (p + 1) + lim_(h rightarrow 0) (sin h)/h`

= (p + 1) + 1

= p + 2  ...(ii)

And f(0⁺) = `lim_(h rightarrow 0) f(0 + h) = (sqrt(h^2 + h) - sqrt(h))/(h^(3//2))`

= `lim_(h rightarrow 0) ((h)^(1/2) [sqrt(h + 1) - 1])/(h(h^(1/2))`

= `lim_(h rightarrow 0) (sqrt(h + 1) - 1)/h xx (sqrt(h + 1) + 1)/(sqrt(h + 1) + 1)`

= `lim_(h rightarrow 0) (h + 1 - 1)/(h(sqrt(h + 1) + 1)`

= `lim_(h rightarrow 0) 1/(sqrt(h + 1) + 1)`

= `1/(1 + 1)`

= `1/2`  ...(iii)

Now, from equation (1),

f(0^–) = f(0) = f(0)⁺  

`\implies` p + 2 = q = `1/2`

`\implies` q = `1/2` and p = `(-3)/2`

∴ (p, q) = `(-3/2, 1/2)`