Calculus I, Chapter 3: The Average Rate of Change and Limits
Previous Section: 3.6 Continuity
At this point we have identified a few methods of solving limits: using a chart to estimate, using a graph and one-sided limits, using continuity to evaluate with substitution. However, this doesn’t tackle all of the problems that can arise with limits. For example, if we have an equation where graphing is difficult and the function is not continuous we are left with the chart method. The chart method is too tedious to be considered effective in many cases, and as humans we like shortcuts.
Using Algebra
A new method can utilize the algebraic skills developed over previous years to help remove the discontinuities in a function. Remove doesn’t mean they don’t exist any longer, but that we compare the function to something similar that doesn’t have the discontinuities that are problematic. Obviously, this is not always possible, but it is particularly helpful in many cases.
Let us look at a few problems to get started.
Factoring a Limit
Expanding a Limit
Conjugates
In the next example, we will trade negative exponents for fractional ones. Radical functions and roots can pose a variety of problems because of the difficulty in cancelling out an exponent. A great method is by multiplying in order to cancel out the ‘problematic’ root. Generally, in a limit problem, there is one root that causes the indeterminate values to occur, so one might identify the where the root is subtracting another value to get 0. Target this part of the problem.
We can use a trick from algebra to get rid of that problematic root. This is called Multiplying by the Conjugate. Recall that the conjugate is the other half of the factored difference of squares. That is, in a^2-b^2=(a-b)(a+b) the (a-b) and (a+b) are a conjugate pair. If we rewrite this with radicals we get the following: a-b^2=(√a-b)(√a+b).
Multiplying by the conjugate means to identify which of the pair is present in your problem, then multiply (both the numerator and denominator) by the other part of the pair. The reason for this is because it cancels out the problematic root that was mentioned before, therefore allowing the problem to be solved.
Review
Limits can be evaluated by factoring, expanding, combining fractions and using conjugates.