Question

If f(x) = `{{:((x - 4)/(|x - 4|) + a",",  "for"  x < 4),(a + b",",  "for"  x = 4  "is continuous at"  x = 4","),((x - 4)/(|x - 4|) + b",",  "for"  x > 4):}`

then ______.

Options

• a = 0, b = 0

• a = 1, b = – 1

• a = – 1, b = 1

• a = 1, b = 1

Answer 1

If f(x) = `{{:((x - 4)/(|x - 4|) + a",",  "for"  x < 4),(a + b",",  "for"  x = 4  "is continuous at"  x = 4","),((x - 4)/(|x - 4|) + b",",  "for"  x > 4):}`

then a = 1, b = – 1.

Explanation:

We have, f(x) = `{{:((x - 4)/(|x - 4|) + a",",  "for"  x < 4),(a + b",",  "for"  x = 4),((x - 4)/(|x - 4|) + b",",  "for"  x > 4):}`

∵ f(x) is continuous at x = 4

∴ `lim_(x rightarrow 4^+) f(x) = lim_(x rightarrow 4^-) f(x)` = f(4)

`lim_(x rightarrow 4^+) f(x) = lim_(x rightarrow 4^+) (x - 4)/(|x - 4|) + b` = b + 1  ...(i)

`lim_(x rightarrow 4^-) f(x) = lim_(x rightarrow 4^-) (x - 4)/(|x - 4|) + a` = a – 1  ...(ii)

f(4) = a + b  ...(iii)

Equating equations (i) and (iii), we get a = 1

EquaJing equations (ii) and (iii), we get b = – 1