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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