# limit

## The Crux of Calculus

Define $\Delta x \equiv x_2 – x_1$, to be consistent with this post. Similarly, define $\Delta y \equiv y_2 – y_1$ and $\Delta z \equiv z_2 – z_1$. The Cartesian coordinates are $x$, $y$, & $z$. In contrast, the spherical coordinates are $r$, $\theta$, & $\phi$. Here, $\phi$ is the azimuthal angle in the $xy$-plane. Next, …

## A Limit Involving the Cosine Function

Now that several limit properties have been proven, it is possible for me to evaluate $\lim_{\alpha \rightarrow 0} \frac{1 – \cos \alpha}{\alpha}$. To do this, I follow the steps in Reference [1]. However, I am going to constrain $\alpha$, in radians, to be greater than or equal to zero, so that I do not …

## Limit of a Sum of Two Functions

In this post, I show that $\lim_{x \rightarrow a} [f(x) + g(x)]$ is equal to $\lim_{x \rightarrow a}f(x) + \lim_{x \rightarrow a} g(x)$ given that $\lim_{x \rightarrow a}f(x) = A$ and $\lim_{x \rightarrow a}g(x) = B$. To do this, I approximately follow the steps in Reference [1]. Objective: Using the definition of a limit, the objective is …