1.3 Limits
You've seen examples of how to find left and right limits. If the left and right limits both agree, then we say the limit exists and equals that common value.
Suppose that
Then we say “the limit as x approaches a of f(x) equals L”, which is written:
An intuitive understanding of limits will suffice for this class. In a more advanced calculus course (Real Analysis) you’ll be introduced to a formal definition of limits. This formal definition of limits can be used to prove the limit properties listed below (and much else!).
Examples 1 & 2: If you look back to examples 1 and 2 on the one-sided limits page, you’ll see that we can conclude that
Click to see these further limit examples:
Properties of Limits
Assume here that
Also assume k and r to be constants (fixed numbers). Then the following theorems hold true:
Proofs of these theorems - Click here!
The above properties can be used to show that for all rational functions Q (assuming the denominator’s limit is nonzero),
The limit properties also imply that functions involving radicals will also satisfy this easy and useful property. The set of all functions that satisfy this particular property can be referred to as the set of functions that are continuous at a. Continuity will be discussed in the following section.
Examples: Click here to see examples involving the above limit properties.
The Squeeze Theorem
Click here for a discussion of the squeeze theorem, applying this concept to prove that
.
Limits Involving Infinity
Limits may involve infinity in various ways. One may take the limit as x goes to positive or negative infinity. Sometimes a limit has x going to a finite value, but the limit ends up going to positive or negative infinity. This is all fairly intuitive once you've seen a few examples.
Click here to see the formal definitions of limits involving infinity.
1.3 Exercises Do all of them! :-)
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