Unit U Big Questions

• What is continuity? What is discontinuity?

  A continuity is a function that is predictable, has no breaks, no holes, and no jumps. You can draw it without lifting your pencil. Some examples of continuous functions are lines and parabolas.

• What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and a value?

  A limit is the intended height of a function. A limit exists when both the left and right reach the same height. The left and right limit must be the same. We can see if the limit does not exist by seeing if our fingers do not meet.

  The difference between a limit and a value is a limit is the intended height whereas the value is the actual height. Sometimes, the limit and value can be the same. 

And other times, they are different. In the picture below, we are looking at lim x->1 f(x)= -2 and f(1)= 0. The limit is the hole and the value is the point above.

• How do we evaluate limits numerically, graphically, and algebraically?

  To evaluate numerically, we use a table. The table is used to find the y-value which refers to the limit of the function.

  To evaluate graphically, we use our graphing calculators. Plug in the function into the y= screen, graph, then trace the value we are looking for. Sometimes a limit does not exist, and we have to explain why. It can be because it is a jump (different left/right), have unbounded behavior (infinite disc.), and oscillating behavior (oscillating disc.).

  To evaluate algebraically, there are a few methods. We can use the direct substitution method which is plugging in the value into the function. If we get 0/0 which is an indeterminate form, then we must use another method. The factoring method is canceling out terms from the numerator and denominator, them using direct substitution to find the limit. The last method is rationalizing/conjugate method where use the conjugate (this can be numerator or denominator, depending on where the radical is). And from there use direct substitution.