Find the value of

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 –

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