Laws of Limits: Theorems and Proofs

According to Theorem 1.1, for any real number $c$, the limit of $x$ as $x$ approaches $c$ is equal to $x$.

Theorem 1.2 states that for any constant $m$ and real number $c$, $lim_{x \to c} m = m$.

If $lim_{x \to c} f(x) = L$ and $k$ is any constant, then $lim_{x \to c} [k + f(x)] = k + L$.

If both $lim_{x \to c} f(x)$ and $lim_{x \to c} g(x)$ exist, then $lim_{x \to c} [f(x) + g(x)] = lim_{x \to c} f(x) + lim_{x \to c} g(x)$.

If $lim_{x \to c} f(x)$ and $lim_{x \to c} g(x)$ both exist, then $lim_{x \to c} [f(x)g(x)] = [lim_{x \to c} f(x)][lim_{x \to c} g(x)]$.

For any functions $f(x)$ and $g(x)$, $lim_{x \to c} [f(x) / g(x)] = [lim_{x \to c} f(x)] / [lim_{x \to c} g(x)]$ always holds true, regardless of the limit of $g(x)$.

If $lim_{x \to c} f(x)$ exists, then $lim_{x \to c} [5f(x)] = 5 lim_{x \to c} f(x)$ according to the scalar multiple rule.

The proof of Theorem 1.1, which states $lim_{x \to c} x = c$, involves setting $\delta = \epsilon^2$ to show that $|x - c| < \epsilon$ whenever $0 < |x - c| < \delta$.

The proof of Theorem 1.2, which addresses the limit of a constant function, uses $\delta = \epsilon$ to show that $|f(x) - k| < \epsilon$ whenever $0 < |x - c| < \delta$.