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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Differentiating the Sine Function

From this post, one type of derivative is $\lim_{\Delta x\rightarrow0^+}\frac{f(a+\Delta x)-f(a)}{\Delta x}\equiv\frac{df(x)}{dx}\big|_{a^+}$ To be consistent with my previous interpretation of $0^+$ in this post, $\Delta x \rightarrow 0^+$ means constraining $\Delta x$ to positive numbers. Next, define…

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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 the Product of Two Functions

In this post, I show that $\lim_{x\rightarrow a}[f(x)g(x)] = \lim_{x\rightarrow a} f(x) \lim_{x\rightarrow a} g(x)$ given that $\lim_{x\rightarrow a} f(x) = A$ and $\lim_{x\rightarrow a} g(x) = B$. To do this, I approximately follow…

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