Combination Theorem for Limits of Functions/Real

Theorem
Let $$f$$ and $$g$$ be a real functions defined on an open interval $$\left({a \, . \, . \, b}\right)$$, except possibly at the point $$c \in \left({a \, . \, . \, b}\right)$$.

Let $$f$$ and $$g$$ tend to the following limits:

$$\lim_{x \to c} f \left({x}\right) = l, \lim_{x \to c} g \left({x}\right) = m$$

Let $$\lambda, \mu \in \mathbb{R}$$ be any real numbers.

Then the following results hold:

Sum of Limits
$$\lim_{x \to c} \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda l + \mu m$$;

Product of Limits
$$\lim_{x \to c} \left({f \left({x}\right) g \left({x}\right)}\right) = l m$$;

Quotient of Limits
$$\lim_{x \to c} \frac {f \left({x}\right)} {g \left({x}\right)} = \frac l m$$, provided that $$m \ne 0$$.

Continuity
If $$f$$ and $$g$$ are continuous on an interval $$\mathbb{I}$$, and $$\lambda, \mu \in \mathbb{R}$$, that:


 * $$\lambda f + \mu g $$ is continuous on $$\mathbb{I}$$;
 * $$f g$$ is continuous on $$\mathbb{I}$$;
 * $$\frac {f} {g}$$ is continuous on $$\mathbb{I}$$.

Proof
These results follow directly from the Combination Theorem for Sequences and Limit of Function by Convergent Sequences, as follows:

Let $$\left \langle {x_n} \right \rangle$$ be any sequence of points of $$\left({a \, . \, . \, b}\right)$$ such that $$\forall n \in \mathbb{N}^*: x_n \ne c$$ and $$\lim_{n \to \infty} x_n = c$$.

By Limit of Function by Convergent Sequences, $$\lim_{n \to \infty} f \left({x_n}\right) = l$$ and $$\lim_{n \to \infty} g \left({x_n}\right) = m$$.

By the Combination Theorem for Sequences:
 * $$\lim_{n \to \infty} \left({\lambda f \left({x_n}\right) + \mu g \left({x_n}\right)}\right) = \lambda l + \mu m$$;
 * $$\lim_{n \to \infty} \left({f \left({x_n}\right) g \left({x_n}\right)}\right) = l m$$;
 * $$\lim_{n \to \infty} \frac {f \left({x_n}\right)} {g \left({x_n}\right)} = \frac l m$$, provided that $$m \ne 0$$.

Applying Limit of Function by Convergent Sequences again, we get:


 * $$\lim_{x \to c} \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda l + \mu m$$;
 * $$\lim_{x \to c} \left({f \left({x}\right) g \left({x}\right)}\right) = l m$$;
 * $$\lim_{x \to c} \frac {f \left({x}\right)} {g \left({x}\right)} = \frac l m$$, provided that $$m \ne 0$$.

It follows trivially from the definitions of continuity at a point and continuity on an interval that, if $$f$$ and $$g$$ are continuous on $$\mathbb{I}$$ that the assertions:


 * $$\lambda f + \mu g $$ is continuous on $$\mathbb{I}$$;
 * $$f g$$ is continuous on $$\mathbb{I}$$;
 * $$\frac {f} {g}$$ is continuous on $$\mathbb{I}$$

are true.