Question

What is the constant "a" such that the function is continuous everywhere?

Find the constant "a" such that the function is continuous everywhere

Answer 1

`a = 4`

Explanation:

We want to start by finding the limit as `x -> 0^-` on `(4sinx)/x`

The limit `lim_(x->0) sinx/x =1` is commonly known and is derived by the squeeze theorem.

So it would only make sense if `lim_(x-> 0) (4sinx)/x = 4(1) = 4`

We want the right hand limit to equal this as well, so we need

`a - 2x = 4`

We know that `x= 0`, thus:

`a - 2(0) = 4`

`a = 4`

We now graph the function to ensure that it's now continuous.

The graph above is clearly continuous, therefore we are correct. Also note that only at `x = 0` could the piecewise function not be continuous, because both `(4sinx)/x` and `4 - 2x` are continuous everywhere else (on all values of `x`).

Hopefully this helps!