Standard limits - single variable calculus

Limits in general refers to the concept of understanding how a function behaves when the input (x) approaches a certain value (a). Standard limits are a specific set of limits that simplifies calculations and are important to learn. In short, all standard limits are limits, but not all limits are standard limits.

Let and be two functions that tend to the same value as approaches some point .

What then is the following limit:

For expressions consisting of a function divided by another function, where the two tend toward the same value as approaches some point, it may not be obvious what value the whole expression tends toward.

Both equations could for example grow larger and larger as tends to infinity, but which one of them grows faster?

This question is often interesting to computer scientists studying time complexity, where they aim to compare the speed of different algorithms.

To help solving this problem, there are a handful of standard limits with known values we can use.

Standard limit 1

In the graph, and both have the value of , but for any exponential function with a base greater than , the exponential will grow faster than the power function in the denominator.

Standard limit 2

Like in the last example, the choice of base for the logarithm and exponent for the polynomial does not matter. The logarithmic curve will always flatten out and be outrun by the polynomial.

Standard limit 3

Notice how the graph of and follow each other around .

Standard limit 4

Standard limit 5

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Get exam problems for old calculus exams divided into chapters

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