Find the values of k so that the function f is continuous at the indicated point f(x)={(k cos x/π−2x, if x ≠ π/2) (3, if x = π/2)

Solution:

The given function is

f(x) = {(k cos x/π − 2x, if x ≠ π/2) (3, if x = π / 2)

The given function f is continuous at x = π / 2,

if f is defined at x = π / 2 and if the value of the f at x = π / 2 equals the limit of f at x = π / 2.

It is evident that f is defined at x = π / 2 and f(π / 2) = 3 

lim_x→π/2 f(x) = lim_x→π/2 k cos x / π − 2x

Put x = π / 2 + h

Then x→π/2

⇒ h→0

⇒ lim_x→π/2 f(x) = lim_x→π/2 k cos x / π − 2x

= lim _h→0 k cos(π/2 + h) / π − 2(π/2 + h)

=k lim _h→0 − sin h − 2h

= k/2 lim _h→0 sin h = k/2.1

= k/2

⇒ lim_{x→π / 2} f(x) = f(π / 2)

⇒ k / 2 = 3

⇒ k = 6

Therefore, the value of k = 6

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 26

Find the values of k so that the function f is continuous at the indicated point f(x)={(k cos x/π−2x, if x ≠ π/2) (3, if x = π/2)

Summary:

The values of k so that the function f is continuous at the indicated point f(x)={(k cos x/π−2x, if x ≠ π/2) (3, if x = π/2) is 6