Limit
The limx→afx=L means that as x becomes closer and closer to a specific value a, the function f(x) will approach L. L is the number that the value approaches. The function never actually reaches this “a” value, so instead we find what it approaches. Let’s take the function fx=x2-1x-1, at x=1, because it equals 00 so we can take the limit to find what y value, x is approaching. I can either take the limit from the right, or the left. For this example I took the limit as it approached from the right. We get the equation: limx→1+x2-1x-1 = L. Now we can create a table of values where it is approaching x from the right. x | 2 | 1.5 | 1.1 | 1.01 | 1.001 | 1.0001 | f(x) | 3 | 2.5 | 2.1 | 2.01 | 2.001 | 2.0001 |
As you can see from the table, as x approaches 1, f(x) approaches 2 but is never actually 2. Since we can’t actually write that f (1)=2 you express it as, limx→1+x2-1x-1=2. The limit does not exist if limx→a-fx≠limx→a+f (x) because in order for the limit to exist, the function must be approaching the “L” value from both the left and the right towards the same number. In the same example above, yes it is possible for this statement f(a)≠L to be true because in in the above example the limit value above is 2 but the function does not exist.

Asymptote
An asymptote is a value that f(x) approaches when x is approaching an “a” value. In the equation limx→a+fx=∞, this is saying that as x becomes sufficiently close to an “a” value, the function f(x) will approach infinity. This means that there is a vertical asymptote at x=a. For example: looking at the graph pictured on the right, as x approaches 0, f(x) is approaching infinity. This limit does not exist because limx→0+1x=∞ and the limx→0-1x=-∞. Technically, the limit doesn’t exist because it is approaching infinity, which isn’t a specific number. These two limits do not head towards the same number or in this case infinity; therefore the limit does not exist. The limx→∞f(x)=L means that there is a horizontal asymptote. For example, in the same graph pictured above, as x approaches infinity, the function f(x)=1x is becoming closer and closer to 0. So we write the limx→∞1x=0. In this case the limit does exist because it is a one-sided limit that is settling on a specific number.

Continuity
For a function f to be continuous at a the limit of f(x) as x approaches a has to equal f(a). There are three conditions that f must satisfy. First, if there is a specific f(a) value for “a”. Meaning that there can’t be a hole at f(a) for a value “a”. Second, the limx→af(x) has to exist. Meaning that as you approach “a” from the left and the right, it has to settle on the same number. Lastly, the limx→afx=f(a). F is discontinuous at x=a if it doesn’t satisfy at least one of the three conditions mentioned above.

The first picture on the left exemplifies the first condition. The second picture exemplifies the second condition because as x approaches 3 from the right its 4 but as x approaches 3 from the left it is approaching 1. Since it does not settle on a single number, this is a point of discontinuity. The third example exemplifies the third condition. This is a picture where the limx→afxdoes not equal f (a). This is because even though the limit at x=2 exists, it is not equal to the value of the function, which is 1.5.

Derivative
The derivative of a function represents the tangent line. When you compute the slope of a tangent line, the instantaneous rate of change of a function, and the instantaneous velocity of an object at x=a all require us to compute the limit . This quantity represents the slope of the tangent line. On a graph it would look like this graph on the right. You can see that the black line represents the tangent line, which is the slope of the derivative of the original function.
There are three situations where a function is not differentiable. First, if the graph has a kink or a cusp then f is not differentiable there. This fails the definition of differentiability because they break the rule that right and left hand limits have to settle on the same number. On the left side of this example, it is approaching infinity, but from the right it is approaching negative infinity. Second, if there is any discontinuity, then f is not differentiable. For example if there is a hole or jump discontinuity. This is not differentiable because in order for it to be differentiable it has to be continuous on an interval [a,b]. Third, if there is a vertical tangent line. Meaning that if the line is completely vertical, it is not differentiable. This is not differentiable because it has an infinite slope