Limits

Limit of Absolute Value

Introduction The limit of an absolute value function often involves determining how the function behaves as the input approaches a particular point, especially around points where the expression inside the absolute value changes sign. Key Concept Often limits involving absolute value do not exist. For example: $$\lim_{x\to0}\dfrac{x}{|x|}.$$ Because of the discontinuity on the graph of…

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Limits of Trig Functions

1 Introduction Oftentimes in calculus, we must compute limits that involve trig functions. For example, $$\lim_{x\to{a}}\sin{x}$$ or the limit of any other expression that contains a trig function. This specific limit asks “what value does $\sin{x}$ approach as $x$ approaches $a$?” Keep in mind that some trig limits do not exist. Because of the oscillation…

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Related Rates Ladder Problem
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Introduction One of the most common related rates problems is the ladder problem, which looks at how a ladder slides down a wall, assuming that the ladder always makes a right triangle with the wall. Let’s see how to solve these sorts of problems by working through a simple example. Example Let’s consider a ladder…

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Limits of Rational Functions

Introduction In calculus, we often need to find the limit of a function as it approaches a certain value. A common scenario involves the limit of a ratio of two functions, which we call rational functions. The general form looks like this: $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ This expression asks: “What value does the fraction approach…

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