Limits: Vital Building Blocks of Calculus

This article is to bring several videos, each focused of a different aspect of limits and their use, into a flow of the concept. 

I am adding more math tutorial videos, so subscribing to my channel will allow you to find these and other videos easily.  There are already other calculus videos available.

It is not possible to fill in a hole in a domain using infinity, but if a function tends to increase, or decrease, without bound from the left and from the right of a missing domain value, the limit is said to be infinity or negative infinity depending on which is being approached. 

Calculus has an important concept, the derivative.  The derivative allows finding the slope of the tangent line of the function at a point.  The theory is that the limit of secant line segments, or chords, approaches the slope of the tangent line as the two points defining the slope of the secant lines become very close.  This results from the points (x + h, f(x + h) and (x, f(x)) becoming very close as h approaches zero. 

A second important concept of calculus is the integral, or the antiderivative.  The integral gives the “area” under a curve, with the understanding that in this sense “area” can be negative.  This is possible because we can partition the area into very narrow rectangles, with the width of the rectangles having a limit of zero.  The rectangles then have their areas added.  The use of a limit is very important here.

Limits form building blocks for both derivatives and integrals, two major concepts of calculus.  While it is possible to avoid using limits in differentiating a function or in integrating a function, limits are at the core of the theory that makes our shortcut formulas possible.

Just as a function can increase or decrease without bound, so can the domain.  What happens as x approaches infinity, or approaches negative infinity?  We cannot approach infinity from both sides, but we can approach negative infinity from the right and infinity from the left.  By taking one-sided limits we can see what happens at the extremes of the function’s domain.

For limits that result in zero over zero, or an undefined expression over an undefined expression, we can often use L’Hopital’s rule to find the limit.  This involves taking the derivatives of the numerator and of the denominator, and seeing if that resolves the difficulty.  It is possible it will not, but it is acceptable to apply L’Hopital’s rule multiple times if necessary.

Limits are not just a way to avoid problems such as divisions by zero, they are important to the theory in two major concepts of calculus. 

Calculus is so much easier if we understand why it works, instead of memorizing many formulas with no idea what makes them valid. 

I am planning to add other articles with more videos embedded.  These will be forthcoming.