## Some elementary properties of limits

Some elementary properties of limits.

1) Suppose for functions f and g, $\lim_{x \rightarrow a}f(x)$ and $\lim_{x \rightarrow a}g(x)$ exist. Then, we have

(a) $\lim_{x \rightarrow a}(f(x)+g(x))=\lim_{x \rightarrow a}f(x)+\lim_{x \rightarrow a}g(x)$

(b) $\lim_{x \rightarrow a}(f(x)g(x))=\lim_{x \rightarrow a}f(x). \lim_{x \rightarrow b}g(x)$

(c) $\lim_{x \rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a}g(x)}$, provided $\lim_{x \rightarrow a}g(x) \neq 0$

(2) Suppose f is continuous at a, then $\lim_{x \rightarrow a}f(x)$ is simply $f(a)$.

(3) Suppose f is defined in a deleted neighbourhood of $x_{0}$, g is defined in a neighbourhood of $x_{0}$ and is continuous. If $f(x)=g(x)$ for x in the deleted neighbourhood of $x_{0}$, then $\lim_{x \rightarrow a}f(x)=g(x)$.

From the above, it is easy to see that

(a) $\lim_{x \rightarrow x_{0}}p(x)=p(x_{0})$ fpr a polynomial p.

(b) $\lim_{x \rightarrow x_{0}}\sin{x}=\sin{x_{0}}$

(c) $\lim_{x \rightarrow x_{0}}\frac{p(x)}{q(x)}=\frac{p(x_{0})}{q(x_{0})}$, for polynomials p, q if $q(x_{0}) \neq 0$.

More later,

Nalin Pithwa

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