# Limits

## Zero to the Zero Power: Indeterminate, or Defined?

Last week we looked at numbers raised to the zero power, as part of our series on oddities of zero. We’ve looked at zero divided by zero in the past, and just recently observed how 0 to the 0 power relates to degree in polynomials, which is part of the motivation for this series. But …

## Subtleties in a Radical Limit Problem

(A new question of the week) I find it interesting to observe the process of problem-solving, particularly for proofs: how we discover a solution initially, and then how we turn that into a final answer. Sometimes we can see the main idea in a flash, but the process of writing it as a formal proof …

## Limits: Recognizing Indeterminate Forms

(A new question of the week) Limits of indeterminate forms like ∞ – ∞ require us first to recognize the form, and then, often, use L’Hôpital’s rule (also called L’Hospital’s rule, as we’ll be seeing it here), or some other method. Today’s question will touch on all stages of this work for three examples, but …

## Limits: What Does “Approach” Mean?

(A new question of the week) We’ve looked at the concept of limit of a function from several perspectives, including why they are needed, and what the definition means. Here we have a more fundamental question, which applies to both functions and sequences: What do we mean when we say a value approaches some number? …

## Limit of sin(x)/x

Last week we looked at some recent questions about limits, where we focused first on what limits are, in terms of graphs or tables, and then on finding them by algebraic simplification. This week, we’ll look at two old questions about a trigonometric limit that can’t be determined that way: sin(x)/x, as x approaches zero.

## Limit Basics: Tables, Graphs, and Simplification

(A new question of the week) I am looking back at recent questions I’ve skipped because, though having useful content, the discussions were cut short. In the two cases we’ll see here, the student who asked a question never read the final answer, perhaps because it went to their spam folder, so the discussion was …

## Separable Differential Equations

(A new question of the week) We received a couple different questions recently about solving differential equations by separation of variables, and why the method is valid. We’ll start with a direct question about it, and then look at an attempt at an alternate perspective using differentials.

## One-sided limits of a composite function

(A new question of the week) A good way to develop a sense of what limits are and how they work comes from working with visual representations of them, in the form of graphs. In particular when the functions are defined by graphs rather than by equations, we have a lot more flexibility in creating …

## Two Inside-Out Limit Problems

(A new question of the week) Limits can be challenging. They can be even more challenging when they require L’Hôpital’s rule or more advanced methods (Maclaurin series), and then are turned inside-out by asking not for the limit itself, but for parameters that will result in a specified limit, or what values of the limit …

## A Geometrical Limit

(A new question of the week) We usually see limits applied to functions in a calculus class. An interesting question from late October deals with a limit in a geometrical construction based on a function. We’ll be seeing how to discover a proof, then several alternative proofs, and finally what the answer means.