Limits and Continuity

Calculus is interested in motion and change of quantities that depend on certain parameters through a rule. For example, the change of velocity of a car with respect to time, change in temperature with respect to the temperature of the surrounding environment, growth of a certain amount of money with respect to interest rate are only a few that comes to mind.

It has two fundamentally significant concepts.The first of these concepts is the derivative, that is finding instantaneous change of a given function at a given point. It corresponds to the problem of finding the slope of a line tangent to a given curve at a given point on the curve. The second problem is concerned with finding the area of a plane region bounded by given curves. The solution of the problem of areas is the subject of the integral concept. 

Suppose that a ball is dropped from the top of a tower of height 200m. In the first t seconds, the distance of the ball travels is given by the equation
y (t)= 4,9t².

Since we do not have any tools for finding the velocity of the ball at the instant t=1, it is not possible, at least for now, to find its velocity at the instant t=1. But we can calculate the average velocity over any time interval. So, we can first calculate the average velocity over a time interval [1, t] and let t get closer and closer to t=1. The distance the ball covers around time t=1 are calculated and presented in the following table.

Now, during short time intervals starting at t=1, we can evaluate the average velocities. These are given in the following table:

If the function f (t) gives the distance of an object moving along the horizontal or vertical coordinate line then instantaneous velocity at the time t₀

The tangent line problem is the problem of finding a line that is tangent to a given curve (circle, graph of any function, etc.) at a given point P on the curve.

We read it as ‘’The limit of f as x approaches x₀ is L ‘’.

Let us note that the expression ‘’x≠x₀’’ in the definition of limit means that even though f may not be defined at the point x₀, we can make the value of f (x) as close as we want to L.

The limit of a function at a given point x₀ determines the behaviour of the function near x₀.

The domain of f is the set all real number except x =2 that is, the function f is not defined at x =2. The function can be simplified by cancelling the common factor (x - 2) of the numerator and denominator for x ≠ 2:

This question is an example of how an algebraic manipulation may be needed to calculate the limit at a given point. In the current case, the function is not defined at x=1 since division by zero is not allowed. However, the numerator of the function takes also the value zero at x =1, and it is not immediate how to calculate the limit. We generally encounter this situation in rational functions where both the numerator and the denominator takes on the value zero at the limit point, say x =x₀. In our case, the function can be simplified by a factorisation in the form:

It is read as the right limit of f is L.

Tt is said that the left limit of f is L.

1. limit of a sum: 

2. limit of a difference:  

3. limit of a product: 

4. limit of a constant multiple: 

5. limit of a quotient: 

The symbols –∞ and +∞ do not represent numbers. They imply unbounded behaviour.

If a function f is continuous at every point of the interval I, where it is defined, we say that f is continuous on the interval I. Also, if f is continuous at every point of its domain of definition, we say that f is a continuous function.