The limit of a function
         The definition of the limit of a function
         Infinite limits
         The limit of a function examples
         Vertical, horizontal and slant (or oblique) asymptotes
The definition of the limit of a function
The limit of a function is a real number L that f (x) approaches as x approaches a given real number a, written
if for any e > 0 there is a  d (e) > 0 such that  | f(x) - L | < e  whenever  | x - a | < d (e).
The definition says, no matter how small a positive number e we take, we can find a positive number d such that, for an arbitrary chosen value of x from
the interval     a - d < x < a + d,
the corresponding function's values lie inside
the interval     L - e < f (x) < L + e,
as shows the right figure.
That is, the function's values can be made arbitrarily close to the number L  by choosing x sufficiently close to a, but not equal to a.
Therefore, the number d, that measures the distance between a point x from the point a on the x-axis, depends on the number e that measures the distance between the point f (x) from the point L on the y-axis.
Example:  Given
           whenever 
A limit is used to examine the behavior of a function near a point but not at the point. The function need not even be defined at the point.
Infinite limits
We write
if  f (x) can be made arbitrarily large by choosing x sufficiently close but not equal to a.
We write
if  f (x) can be made arbitrarily large negative by choosing x sufficiently close but not equal to a.
The limit of a function examples
Example:  Evaluate the following limits;
Solution:  a)  As x tends to minus infinity f (x) gets closer and closer to 0.
As x tends to plus infinity f (x) gets closer and closer to 0. Therefore,
b)  As x tends to 0 from the left  f(x) gets larger in negative sense.
As x tends to 0 from the right  f (x) gets larger in positive sense. Therefore,
Vertical, horizontal and slant (or oblique) asymptotes
If a point (x, y) moves along a curve  f (x) and then at least one of its coordinates tends to infinity, while the distance between the point and a line tends to zero then, the line is called the asymptote of the curve.
Vertical asymptote
If there exists a number a such that
then the line  x = a  is the vertical asymptote.
Horizontal asymptote
If there exists a number c such that
then the line  y = c  is the horizontal asymptote.
Slant or oblique asymptote
If there exist limits
then, a line  y = mx + is the slant asymptote of the function  f (x).
Example:  Find the vertical and the horizontal
asymptote of the function
Solution:  Since,
then  x = 1  is the vertical asymptote.
Since,  
then  y = 2  is the horizontal asymptote.
Example:  Calculate asymptotes and sketch the graph of the function
Solution:  By equating the numerator with zero and solving for x we find the x-intercepts,  
x2 - x - 2 = (x + 1)(x - 2) = 0, 
x1 = -and  x2 = 2.
We calculate  f (0) to find the y-intercept,
 f (0) = 2/3.
By equating the denominator with zero and solving for x we find the vertical asymptote,
x = 3.
Let calculate following limits
to find the slant asymptote  y = mx + c.
Therefore, the line  y = x + 2 is the slant asymptote of the given function.
Calculus contents B
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