Question

If f(x) = `{{:((3 sin πx)/(5x),",", x ≠ 0),(2k,",", x = 0):}`

is continuous at x = 0, then the value of k is ______.

Options

• `π/10`

• `(3π)/10`

• `(3π)/2`

• `(3π)/5`

Answer 1

If f(x) = `{{:((3 sin πx)/(5x),",", x ≠ 0),(2k,",", x = 0):}`

is continuous at x = 0, then the value of k is `underlinebb((3π)/10)`.

Explanation:

Given, f(x) = `{{:((3 sin πx)/(5x),",", x ≠ 0),(2k,",", x = 0):}` 

Now, `lim_(x rightarrow 0) f(x) = lim_(x rightarrow 0) ((3 sin πx)/(5x))`

= `3/5 lim_(x rightarrow 0) (sin  (πx)/(πx)) xx π`

= `3/5 xx 1 xx π`

= `3/5 π`

Also, f(0) = 2k

Since, f(x) is continuous at x = 0.

∴ f(0) = `lim_(x rightarrow 0) f(x)`

`\implies` 2k = `3/5 π`

`\implies` k = `(3π)/10`