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Oftentimes when you study continuity, you’ll be presented with problems in which you’re asked to solve for the value of some constant within a piecewise-defined function that will make the value of that function continuous.
Basically what’s happening is you’re being given equations that define the function on either side of a particular value of x. In other words, they’re giving you the left- and right-hand limits. In order for a function to be continuous at a point, the left- and right-hand limits must be equal. So you’re trying to solve for the value of the constant that will make the pieces equal to each other and the “break point”.
In order to solve a problem like this one, you want to plug in the x-value of the “break point” then take the left-hand limit of the piece that defines the function on the left and the right-hand limit of the piece that defines the function on the right, and then set the resulting pieces equal to one another.
The value you find for the constant is the value that forces the left- and right-hand limits to come together and be equal at the “break point”, which would make the general limit exist at that point, and therefore make the function continuous there.
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