Find the value of k, for which

is continuous at x = 0.

OR

If x = a cos³ θ and y = a sin³ θ, then find the value of at

Given,

…(1)

As, f(x) is continuous at x = 0

∴ Left hand limit(at x = 0) = RHL(at x = 0) = f(0)

⇒

To find the value of k we can consider LHL = f(0), as calculation will be easier and fast. You can take any other consideration.

∴

⇒ {using equation 1}

⇒

⇒

As limit is taking 0/0 form so we need to rationalize the expression.

∴

Using (a+b)(a-b) = a² – b² and applying algebra of limits

We have-

⇒

⇒

⇒

⇒ -k = 1

∴ k = -1

OR

Given,

As x = acos³ θ …(1)

Differentiating x w.r.t θ we get-

⇒ {using chain rule}

⇒ …(2)

Similarly we have,

y = a sin³ θ …(3)

Differentiating y w.r.t θ we get-

⇒

⇒ {using chain rule}

⇒ …(4)

By chain rule we can write that:

∴ {from 2 and 4}

⇒

⇒

Again differentiating w.r.t x we get –

⇒

From equation 2 we have –

⇒

∴

∴