If f ( x ) = { a | x | + x 2 - 2 ( sin | x | ( cos | x | ) x , x 0 b , x = 0 is continuous at x=0, then a+b is equal to [2026]

Question

If $f (x)$ $=$ ${\begin{matrix} \frac{a | x | + x^{2} - 2 (\sin | x |) (\cos | x |)}{x} & , x \neq 0 \\ b & , x = 0 \end{matrix}$

is continuous at $x = 0$ , then $a + b$ is equal to [2026]

Smart Achievers · Accepted Answer

(3)

$f (x)$ ${\begin{matrix} \frac{a | x | + x^{2} - 2 (\sin | x |) (\cos | x |)}{x} & , x \neq 0 \\ b & , x = 0 \end{matrix}$ ${\begin{matrix} \frac{a | x | + x^{2} - 2 \sin | x | \cos | x |}{x}, & x \neq 0 \\ b, & x = 0 \end{matrix}$

$For continuity:$

$\lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{+}} f (x) = f (0)$

$\lim_{x \to 0^{-}} \frac{a h + h^{2} - 2 (\sin h) \cos h}{- h}$

$= \lim_{x \to 0^{+}} \frac{a h + h^{2} - 2 (\sin h) \cos h}{h}$

$or - a + 2 = a - 2 = b$

$2 a = 4$

$a = 2, b = 0$

$∴$ $a + b = 2$

Solution

Q.
If $f (x)$ $=$ ${\begin{matrix} \frac{a | x | + x^{2} - 2 (\sin | x |) (\cos | x |)}{x} & , x \neq 0 \\ b & , x = 0 \end{matrix}$

is continuous at $x = 0$ , then $a + b$ is equal to [2026]

1 0

2 4

3 2

4 1

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Solution

Q. If f(x)={a|x|+x2-2(sin|x|)(cos|x|)x, x≠0b, x=0 is continuous at x=0, then a+b is equal to [2026]

1 0

2 4

3 2

4 1

Ans. (3) f(x)={a|x|+x2-2sin|x|cos|x|x,x≠0b,x=0 For continuity: limx→0-f(x)=limx→0+f(x)=f(0) limx→0-ah+h2-2(sinh)cosh-h =limx→0+ah+h2-2(sinh)coshh or -a+2=a-2=b 2a=4 a=2, b=0 ∴ a+b=2

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Q.
If $f (x)$ $=$ ${\begin{matrix} \frac{a | x | + x^{2} - 2 (\sin | x |) (\cos | x |)}{x} & , x \neq 0 \\ b & , x = 0 \end{matrix}$

is continuous at $x = 0$ , then $a + b$ is equal to [2026]