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Section 3.4 How to evaluate limits

Well, now that we know everything about limits, why not evaluate them? Let us now study various techniques to evaluate limits!

Subsection 3.4.1 Instructional video

Subsection 3.4.2 Key concepts

Concept 3.4.1. Limit laws.

Assume that limxaf(x) and limxag(x) exist. Then:

  • Limit of a constant: limxac=c for any cR.
  • Limit of x: limxax=a.
  • Sum rule: limxa(f(x)±g(x))=limxaf(x)±limxag(x).
  • Product rule: limxa(f(x)g(x))=(limxaf(x))(limxag(x)).
  • Quotient rule: limxaf(x)g(x)=limxaf(x)limxag(x) if limxag(x)0.
  • Root rule: limxa(f(x))1/n=(limxaf(x))1/n, where n is a positive integer. For n even we require that limxaf(x)>0 so that the root is real and the limit is well defined.
Concept 3.4.2. Simplifying a function.

If f(x)=g(x) for xa, then

limxaf(x)=limxag(x),

provided the limits exist. In practice, what this means is that when you are evaluating limxaf(x), then you can manipulate f(x) by simplifying it using things like factorization, rationalization, common denominator, etc., assuming that xa, and it will not change the limit.

Concept 3.4.3. Flowchart.

This flowchart is for calculating the limit

limxaf(x)g(x)

for simple enough functions f(x) and g(x) (not piecewise-defined, no absolute values, etc.) such that both f(a) and g(a) are finite. This is the form of many functions that we will encounter for the time being. If your function is not of this form, you may first need to use algebraic techniques, such as common denominator, to transform it into this form.

Figure 3.4.4. Flowchart for evaluating limits
Concept 3.4.5. Squeeze theorem.

First, if f(x)g(x) for all x near a (except possibly at x=a), and if the limits of both functions exist as x approaches a, then

limxaf(x)limxag(x).

The Squeeze theorem states that, if g(x)f(x)h(x) for all x near a (except possibly at x=a), and if

limxag(x)=limxah(x)=L,thenlimxaf(x)=L.

Further readings 3.4.3 Further readings