This Calculus Problem of the Week focuses on computing limits. We guarantee that you will have exam questions very much like these, and so we encourage you to practice many of the various types so you’ll be fully ready. Detailed Solutions (a) Use Factoring to Find a Limit \(\displaystyle{\lim_{x \to 2}\dfrac{x^2 + 3x-10}{x-2} = ?}… [read more]

Are you working with Limit at Infinity problems in Calculus? Let’s break ’em down and develop your understanding so you can solve them routinely for yourself. I.   How to think about “going to infinity” • When you see $\displaystyle{\lim_{x \to \, \infty}}$, think: “The limit as x grows and grows, and Grows, and GROWS, …,… [read more]

Need to solve Optimization problems in Calculus? Let’s break ’em down and develop a strategy that you can use to solve them routinely for yourself. Overview Optimization problems will always ask you to maximize or minimize some quantity, having described the situation using words (instead of immediately giving you a function to max/minimize). Typical phrases… [read more]

In Part 1 of this series, we illustrated three of the most common tactics you must know to use in order to be able to solve limit problems in Calculus: (See Part 1 for details on those.) In this post, we’re going to look at two other tactics you’ll frequently need to invoke. I. Tactic… [read more]

If you’re like many Calculus students, you understand the idea of limits, but may be having trouble solving limit problems in your homework, especially when you initially find “0 divided by 0.” In this post, we’ll show you the techniques you must know in order to solve these types of problems. I. The idea of… [read more]

A student recently wrote to ask if we’d help solve a common related rates problem about water draining from a cylindrical tank. The problem was something like this: Cylinder Drains Water. A cylinder filled with water has a 3.0-foot radius and 10-foot height. It is drained such that the depth of the water is decreasing… [read more]

Are you having trouble solving Calculus limit problems, even though you understand the concept? In this post we explain three approaches you’ll use again and again, especially in problems where you initially get “0/0.” I. Factoring You’ll use this approach most often. For example, consider the problem $$\lim_{x \to 4}\dfrac{x^2 – 16}{x-4} = ?$$ If… [read more]

In our last post, we developed four steps to solve any related rates problem. We introduced three examples to illustrate the basic ideas, and solved two of them there. As promised, we’ll solve the third here. Water Leaving a Cone Example Here’s the problem statement, now with some additional details about the cone itself and… [read more]

Almost 90% of respondents find this blog post “very” or “extremely useful.” We hope you will too! If you’re working on a related rates problem and need help, please let us know on our Forum and we’ll do our best to assist. Are you having trouble with Related Rates problems in Calculus? Let’s break ’em… [read more]

I created Matheno for one simple reason: to help students do well in their college-level science and math courses. As a former Physics faculty member at UC Berkeley, I’ve worked with many beginning students from diverse backgrounds. Most work hard to do well, and many succeed greatly. But others, despite the same amounts of time… [read more]