## 1-5: The Limit of a Function

Definition (1): Suppose  $f(x)$  is defined when  $x$  is near the number  $a$  (This means that  $f$  is defined on some open interval that contains  $a$  , except possibly at  $a$  itself.) Then we write

$\lim_{x\to a}f(x)=\mathbf{L}$

and say “the limit of  $f(x)$  , as  $f(x)$  approaches  $a$  , equals   $\mathbf{L}$

if we can make the values of  $f(x)$   arbitrarily close to  $\mathbf{L}$  (as close to  $\mathbf{L}$  as we like) by taking  $x$  to be sufficiently close to  $a$  (on either side of  $a$  ) but not equal to  $a$

One- Sided Limits.

Definition (2): We write

$\mathbf{\lim_{x\to a^{-}}f(x)=L}$

and say the left- hand limit of  $\mathbf{f(x)}$  as  $x\mathbf{}$  approaches  $a\mathbf{}$  (or the limit of  $f(x\mathbf{})$  as  $x$  approaches  $a$  from the left ) is equal to  $L$  if we can make the values of  $f(x)$  arbitrarily close to   $L$  by taking   $x$  to be sufficiently close to   $a$  and  $x$  less than  $a$

Notice:that Definition (2) differs from Definition (1) only in that we require  $x$  to be less than  $a$  Similarly, if we require that  $x$  be greater than  $a$  we get “the right-hand limit of  $f(x)$  as  $x$  approaches  $a$  is equal to  $L$ ” and we write

$\lim_{x\to a^{+}}f(x)=L$

Thus the symbol “$x\rightarrow a^{+}$” means that wa consider only  $x> a$

3.  $\lim_{x\to a}f(x)=L$  if and only if  $\lim_{x\to a^{-}}f(x)=L$  and  $\lim_{x\to a^{+}}f(x)=L$

Infinite Limits

Definition (4): Let  $f$  be a function defined on both sides of  $a$  except possibly at  $a$  itself. Then

$\lim_{x\to a}f(x)=\infty$

means that the values of  $f(x)$  can be made arbitrarily large (as large as we please) by taking  $x$  sufficiently close to  $a$  but not equal to  $a$

Definition (5): Let  $f$  be defined on both sides of  $a$  except possibly at  $a$  itself. Then

$\lim_{x\to a}f(x)=-\infty$

means that the values of  $f(x)$  can be made arbitrarily large negative by taking  $x$  sufficiently close to  $a$  but not equal to  $a$

Definition (6): the line  $x=a$  is called vertical asymptote of the curve  $y=f(x)$  if at least one of following statements is true:

$\lim_{x\to a}f(x)=\infty$    $\lim_{x\to a^{-}}f(x)=\infty$    $\lim_{x\to a^{+}}f(x)=\infty$

$\lim_{x\to a}f(x)=-\infty$    $\lim_{x\to a^{-}}f(x)=-\infty$    $\lim_{x\to a^{+}}f(x)=-\infty$

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