Find the value of k so that the function f is continuous at the indicated point:

Given,

…(1)

We need to find the value of k such that f(x) is continuous at x = 0.

A function f(x) is said to be continuous at x = c if,

Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).

Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)

Now, let’s assume that f(x) is continuous at x = 0.

∴

As we have to find k so pick out a combination so that we get k in our equation.

In this question we take LHL = f(0)

∴

⇒ {using eqn 1}

⇒ -1

As we can’t find the limit directly because it is taking 0/0 form.

So, we will rationalize it.

⇒ -1

Using (a+b)(a-b) = a² – b² , we have –

-1

⇒ -1

⇒ -1

⇒ = -1

∴ 2k/2 = -1

∴ k = -1