If fx-sin-2x-sin xx - x0- is continuous at x-0 then a-2b is equal to -

F(x) is continuous at x=0 ∴x→ 0^ lim f(x)=b=x→ 0^+lim f(x) b=h→ 0lim f(0+h)=h→ 0lim(h+3h^2)^1/3 h^1/3h^4/3 ⇒ b=h→ 0lim(1+3h)^1/3 1h ⇒ b=h→ 0lim13(1+3h)^ 2/3×3 ...

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Byju's Answer

Standard XI

Mathematics

Continuity of a Function

If fx=sinα+2x...

Question

If $f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (α + 2) x + sin x}{x}, & x < 0 b, & x = 0 \frac{(x + 3 x^{2})^{\frac{1}{3}} - x^{\frac{1}{3}}}{x^{\frac{4}{3}}}, & x > 0 \end{matrix}$ is continuous at $x = 0$ then $a + 2 b$ is equal to :

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Solution

The correct option is C $0$
$f (x)$ is continuous at $x = 0$
$∴ lim x \to 0^{-} f (x) = b = lim x \to 0^{+} f (x) b = lim h \to 0 f (0 + h) = lim h \to 0 \frac{(h + 3 h^{2})^{\frac{1}{3}} - h^{\frac{1}{3}}}{h^{\frac{4}{3}}}$
$\Rightarrow b = lim h \to 0 \frac{(1 + 3 h)^{\frac{1}{3}} - 1}{h}$
$\Rightarrow b = lim h \to 0 \frac{1}{3} (1 + 3 h)^{- \frac{2}{3}} \times 3$
$\Rightarrow b = 1$

$lim x \to 0^{-} f (x) = 1$
$\Rightarrow lim h \to 0 \frac{sin ((α + 2) (- h)) + sin (- h)}{- h} = 1$
$\Rightarrow a + 3 = 1 \Rightarrow a = - 2$
$\Rightarrow a + 2 b = 0$

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If f(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩sin(α+2)x+sinxx ,x<0b ,x=0(x+3x2)13−x13x43 ,x>0 is continuous at x=0 then a+2b is equal to :

The correct option is C 0f(x) is continuous at x=0 ∴limx→0−f(x)=b=limx→0+f(x)b=limh→0f(0+h)=limh→0(h+3h2)13−h13h43 ⇒b=limh→0(1+3h)13−1h ⇒b=limh→013(1+3h)−23×3 ⇒b=1 limx→0−f(x)=1 ⇒limh→0sin((α+2)(−h))+sin(−h)−h=1 ⇒a+3=1⇒a=−2 ⇒a+2b=0