Find a value for a so that the following function is continuous at x = 0: f(x) ={ (x^2 + x sin(x))/x^2 for x /= 0, a for x = 0}

Question

Not sure what to do with this problem. Its a piecewise function.

 

 

Find a value for a so that the following function is continuous at x = 0: f(x) ={ (x^2 + x sin(x))/x^2 for x /= 0, a for x = 0}

Answer 1

Another way to solve this is to apply L'Hopital's rule twice to find the limit as x→0.

 

First application:

 

Limit as x→0 of 

 

(2x + sinx + xcosx) / (2x) is still indeterminate.

 

Second application:

Limit as x→0 of

(2 + 2cosx - xsinx) / 2 = (2 + 2)/2 = 4/2 = 2

Answer 2

Let f(x) = (x²+xsinx)/x², we need lim_x→0f(x) = f(0)

 

lim_x→0f(x) = lim_x→0[(x²/x²)+(xsinx)/x²]

 

               = lim_x→0[1 + (sinx)/x] = 1 + 1 = 2

 

So, f(0) = a = 2 will guarantee the continuity of f(x) when x = 0.