Limit of Difference¶

Suppose that $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist. By noticing that $f(x)-g(x) = f(x) + (-1)g(x)$, we get $$ \begin{align} \lim_{x \to a} (f(x)-g(x)) &= \lim_{x \to a} \Bigl( f(x) + (-1)g(x) \Bigr) \\ &= \lim_{x \to a} f(x) + \lim_{x \to a} \Bigl( (-1)g(x) \Bigr) \\ &= \lim_{x \to a} f(x) + \underbrace{\Bigl( \lim_{x \to a} (-1) \Bigr)}_{-1}\Bigl( \lim_{x \to a} g(x) \Bigr) \\ &= \lim_{x \to a} f(x) - \lim_{x \to a} g(x). \end{align} $$ Here we used most of the results that we have proved so far:

• Limit of sum: we can take the limits of $f(x)$ and $(-1)g(x)$ separately.
• Limit of product: we can take the limits of $-1$ and $g(x)$ separately.
• Limit of a constant: the limit of $-1$ is $-1$.