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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.
Solution
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)
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