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 $x$, $y$,…

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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…

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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)…

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