Finding continuity given a formula

Question

The function 𝑓 is given by the formula: f(x)=2x/(x^2+6x) when x<0 and by the formula f(x)=cos(x)+a when x≥0. What value must be chosen for 𝑎a in order to make this function continuous at 𝑥=0 ?

Answer 1

Choose a so that the one sided limits of f(x) as x approaches 0 are equal.

As x approaches 0 from the left, we have f(x) = 2x / (x² + 6x) = 2x / [x(x + 6)] = 2 / (x + 6). So the limit of f(x) as x approaches 0 from the left is 2 / (0 + 6) = 1/3

As x approaches 0 from the right, the limit is cos0 + a = 1 + a.

So, we want 1 + a = 1/3

Therefore, a = -2/3

Answer 2

desmos.com/calculator/txjpecpiga

Visit the above graph and click the link to "Make f(x) continuous"