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Question: If f(x) = \(\left( \frac{x - 2}{2} \right) - \frac{\pi}{6}\) is continuous at x = 3, then λ =...

If f(x) = (x−22)−π6\left( \frac{x - 2}{2} \right) - \frac{\pi}{6} is continuous at x = 3, then λ =

A

4

B

3

C

2

D

1

Answer

1

Explanation

Solution

L.H.L. at x = 3, lim⁡x→3−f(x)=lim⁡x→3−(x+λ)\lim _ { x \rightarrow 3 ^ { - } } f ( x ) = \lim _ { x \rightarrow 3 ^ { - } } ( x + \lambda )

= lim⁡h→0(3−h+λ)\lim _ { h \rightarrow 0 } ( 3 - h + \lambda )=3+λ3 + \lambda …..(i)

R.H.L. at x = 3, lim⁡x→3+f(x)=lim⁡x→3+(3x−5)\lim _ { x \rightarrow 3 ^ { + } } f ( x ) = \lim _ { x \rightarrow 3 ^ { + } } ( 3 x - 5 )

= lim⁡h→0{3(3+h)−5}\lim _ { h \rightarrow 0 } \{ 3 ( 3 + h ) - 5 \} = 4 …..(ii)

Value of function f(3)=4f ( 3 ) = 4 …..(iii)

For continuity at x = 3

Limit of function = value of function 3 + λ = 4 ⇒ λ = 1.