Question

The function `f(x) = {{:((e^(3x) - e^(-5x))/x",", if x ≠ 0),(k, if x = 0):}` is continuous at x = 0 for the value of k, as ______.

Options

• 3

• 5

• 2

• 8

Answer 1

The function `f(x) = {{:((e^(3x) - e^(-5x))/x",", if x ≠ 0),(k, if x = 0):}` is continuous at x = 0 for the value of k, as 8.

Explanation:

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

LHL= RHL = f(0) or LHL = RHL = k

Now, LHL = `lim_(h rightarrow 0) (e^(3(0 - h)) - e^(-5(0 - h)))/(0 - h)`

= `lim_(h rightarrow 0) (e^(-3h) - e^(5h))/(-h)`

= `lim_(h rightarrow 0) ((e^(-3h) - 1)/(-h)) + lim_(h rightarrow 0) ((e^(5h) - 1)/h)`

= `3lim_(h rightarrow 0) ((e^(-3h) - 1)/(-3h)) + 5lim_(h rightarrow 0) ((e^(5h) - 1)/(5h))`

= 3 × 1 + 5 × 1 

= 8

Thus, k = 8.