Limits, Continuity, and Trigonometric Limits

Limits, Continuity, and Trigonometric Limits...
Sine and Cosine are ratios defined in terms of the acute angle of a right-angled triangle and the sides of the triangle.
Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, ... In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function. ..... (the left-hand limit does not exist due to the oscillatory nature of the sine .
From this example we can get a quick “working” definition of continuity. A function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil. The graph in the last example has only two discontinuities since there are only two places where we would have to pick up our pencil in sketching it.

In other words, a function is continuous if its graph has no holes or breaks in it.



For many functions it’s easy to determine where it won’t be continuous. Functions won’t be continuous where we have things like division by zero or logarithms of zero. Let’s take a quick look at an example of determining where a function is not continuous.