-fracsin a-1 x-2 sin- xx- if w0 -endmatrix -right- is continuous at x -0- then find the values of a and b-

LHL (at x = 0) : lim_x → 0^ sin(a+1)x+2sin x/x=lim_x → 0^ (sin(a+1)x/(a+1)x ) × (a+1)+2 ( sin x/x ) =1 × (a+1)+2 × 1=a + 3 [As x → 0 ∴ (a+1)x → 0 RH ...

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Byju's Answer

Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 2

⟨fracsin a+1 ...

Question

$I f f (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{s i n (a + 1) x + 2 s i n x}{x} & i f x < 0 2, & i f x = 0 \frac{\sqrt{1 + b x} - 1}{x}, & i f x > 0 \end{matrix}$ is continuous at x = 0, then find the values of a and b.

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Solution

LHL (at x = 0) : ${lim}_{x \to 0^{-}} \frac{s i n (a + 1) x + 2 s i n x}{x} = {lim}_{x \to 0^{-}} (\frac{s i n (a + 1) x}{(a + 1) x}) \times (a + 1) + 2 (\frac{s i n x}{x}) = 1 \times (a + 1) + 2 \times 1 = a + 3 [A s x \to 0 ∴ (a + 1) x \to 0 R H L (a t x = 0) : {lim}_{x \to 0^{+}} \frac{\sqrt{1 + b x} - 1}{x} = {lim}_{x \to 0^{+}} b \times \frac{1}{\sqrt{1 + b x} + 1} = \frac{b}{2}$

Also, f(0) = 2

As f(x) is continuous at x = 0 so, a + 3 = 2 = $\frac{b}{2} ∴ a = - 1, b = 4.$

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If f(x)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩sin(a+1)x+2sin xxif x<02,if x=0√1+bx−1x,if x>0 is continuous at x = 0, then find the values of a and b.

LHL (at x = 0) : limx→0−sin(a+1)x+2sin xx=limx→0−(sin(a+1)x(a+1)x)×(a+1)+2(sin xx)=1×(a+1)+2×1=a+3 [As x→0∴(a+1)x→0RHL(at x=0):limx→0+√1+bx−1x=limx→0+b×1√1+bx+1=b2 Also, f(0) = 2 As f(x) is continuous at x = 0 so, a + 3 = 2 = b2 ∴a=−1, b=4.

$I f f (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{s i n (a + 1) x + 2 s i n x}{x} & i f x < 0 2, & i f x = 0 \frac{\sqrt{1 + b x} - 1}{x}, & i f x > 0 \end{matrix}$ is continuous at x = 0, then find the values of a and b.