Question

Determine the values of a, b, c for which the function f(x) = `{((sin(a + 1)x + sin x)/x, "for"   x < 0),(x, "for"  x = 0),((sqrt(x + bx^2) - sqrtx)/(bx^(3"/"2)), "for"  x > 0):}` is continuous at x = 0.

Answer 1

Given:

f(x) is continuous at x = 0

For f(x) to be continuous at x = 0, f(0)^- = f(0)⁺ = f(0)

LHL = f(0)^- = `lim_(x->0) (sin (a + 1)x + sinx)/x`

`=> lim_(h->0)(sin (a + h)h + sinh)/h`

`=> lim_(h->0)(sin (a + 1)h)/h + lim_(h->0)sinh/h`

`=> lim_(h->0)(sin(a + 1)h)/h xx ((a + 1))/((a + 1)) + lim_(h->0)sinh/h`

`=> lim_(h->0)(sin (a + 1)h)/((a + 1)) xx ((a + 1))/1 + lim_(h->0)sinh/h`

`lim_(h->0)(sin(a + 1)h)/((a + 1)h) = 1`

`lim_(h->0)sinh/h = 1`

⇒ 1 × (a + 1) + 1

⇒ (a + 1) + 1

f(0)^- ⇒ a + 2       ...(1)

RHL = f(0⁺) = `lim_(x->0)(sqrt(x + bx^2) - sqrtx)/(bx^(3/2))`

`=> lim_(x->0)(sqrt(x + bx^2)- sqrtx)/(bx^(3/2))`

`=> lim_(h->0)(sqrt(h + bh^2) - sqrth)/(bh^(3/2))`

`=> lim_(h->0)(sqrt(h + bh^2) - sqrth)/(b xx h xx h^(1/2))`

`=> lim_(h->0) (sqrt(h + bh^2)-sqrth)/(b xx h xx sqrth)`

`=> lim_(h ->0)(sqrt(h(1 + bh))- sqrth)/(b xx h xx sqrth)`

`=> lim_(h ->0)(sqrth(sqrt(1 + bh))- sqrt1)/(bh xx sqrth)`

`=> lim_(h->0)((sqrt(1 + bh))- sqrt1)/(bh)`

Take the complex conjugate of 

`(sqrt(1 + bh)- sqrt 1)`,

i.e, `(sqrt(1 + bh)- sqrt 1)` and multiply it with numerator and denominator

`=> lim_(h->0)((sqrt(1 + bh))- sqrt1)/(bh) xx ((sqrt(1 + bh)) + sqrt1)/((sqrt(1 + bh)) + sqrt1)`

`lim_(h->0) ((sqrt(1 + bh))^2 - (sqrt1)^2)/(bh)`

∴ (a + b)(a − b) = a² − b²

`=> lim_(h->0)((1 + bh - 1))/(bh(sqrt(1 + bh))+ sqrt1)`

`=> lim_(h->0)((bh))/((sqrt(1 + bh)) sqrt1)`

`=> 1/((sqrt(1 + b xx 0)) + sqrt1)`

`=> 1/(1 + 1)`

f(0)⁺ = `1/2`   ...(2)

since, f(x) is continuous at x = 0, From (1) & (2), we get,

⇒ a + 2 = `1/2`

⇒ a = `1/2 - 2`

⇒ a = `(-3)/2`

Also, 

f(0)^- = f(0)⁺ = f(0)

⇒ f(0) = c

⇒ c = a + 2 = `1/2`

⇒ c = `1/2`

So the values of a = `(-3)/2,` c = `1/2` and b = R-{0}(any real number except 0)