Definition of a Limit
The Intuitive Approach
The precise ("formal") definition of limit carefully defines the ideas that we have already been using graphically and intuitively. The following side–by–side columns show some of the phrases we have been using to describe limits, and those phrases, particularly
the last ones, provide the basis to building the definition of limit.
Let's examine what the last phrase ("we can.."). means for the Particular Limit.
Example 1: We know . Show that we can guarantee that the values of are as close to 5 as we want by starting with values of sufficiently close to 3.
(a) What values of guarantee that is within unit of ? (Fig. 1a)
Solution: "within unit of " means between and , so the question can be rephrased as "for what values of is between 4 and ? We want to know which values of put the values of into the shaded band in Fig. 1a. The algebraic process is straightforward: solve for to get and . We can restate this result as follows: "If is within units of 3, then is within unit of ". (Fig. 1b)
Any smaller distance also satisfies the guarantee: e.g., "If is within units of 3, then is within 1 unit of 5". (Fig. 1c)
(b) What values of guarantee the is within units of ? (Fig. 2a)
Solution: "within units of 5" means between and , so the question can be rephrased as "for what values of is between and 5.2: ?" Solving for , we get and . "If is within units of 3, then is within units of " (Fig. 2b) Any smaller distance also satisfies the guarantee.
Rather than redoing these calculations for every possible distance from 5, we can do the work once, generally:
(c) What values of guarantee that is within units of 5? (Fig. 3a)
Solution: "within unit of 5" means between and , so the question is "for what values of is between and ?" Solving for get and . "If is within units of 3, then is within units of 5". (Fig. 3b) Any smaller distance also satisfies the guarantee.
Part (c) of Example 1 illustrates a little of the power of general solutions in mathematics. Rather than doing a new set of similar calculations every time someone demands that be within some given distance of 5, we did the calculations once. And then we can respond for any given distance. For the question "What values of guarantee that is within and units of 5?", we can answer "If is within and units of 3".
Practice 1: . What values of guarantee that is within
(a) 1 unit of 3?
(b) 0.08 units of 3?
(c) E units of 3? (Fig. 4)
The same ideas work even if the graphs of the functions are not straight lines, but the calculations are more complicated.
Example 2: (a) What values of guarantee that is within unit of ? (b) Within units of ?
(Fig. 5a) State each answer in the form "If is within _____ units of , then is within (or unit of ".
Solution; (a) If is within unit of , then so
or . The interval containing these values extends from units to the left of to units to the right of 2. Since we want to specify a single distance on each side of , we can pick the smaller of the two distances, . (Fig. 5b)
"If is within units of , then is within unit of ".
(b) Similarly, if is within units of 4, then so or . The interval containing these values extends from units to the left of 2 to units to the right of 2. Again picking the smaller of the two distances, "If is within units of , then is within unit of ".
The situation in Example 2 of different distances on the left and right sides is very common, and we always pick our single distance to be the smaller of the distances to the left and right. By using the smaller distance, we can be certain that if is within that smaller distance on either side, then the value of is within the specified distance of the value of the limit.
Practice 2: . What values of guarantee that is within 1 unit of 3? Within units of 3? (Fig. 6) State each answer in the form.
"If is ______ within units of 2, then is within 1 (or 0.2) unit of 4".
The same ideas can also be used when the function and the specified distance are given graphically, and in that case we can give the answer graphically.
Example 3: In Fig. 7, . What values of guarantee that is within units (given graphically) of 3? State your answer in the form "If is within _____ (show a distance D graphically) of 2, then is within units of 3".
Solution: The solution process requires several steps as illustrated in Fig. 8:
i. Use the given distance to find the values and on the -axis.
ii. Sketch the horizontal band which has its lower edge at and its upper edge at .
iii. Find the first locations to the right and left of where the graph of crosses the lines and and at these locations draw vertical lines to the -axis.
iv. On the -axis, graphically determine the distance from to the vertical line on the left (labeled and from 2 to the vertical line on the right (labeled .
v. Let the length be the smaller of the lengths and .
Practice 3: In Fig. 9, . What of guarantee that is within E units of ?