ALGEBRA - solved problems
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).
Continuous function
A real function  y = f (x)  is continuous at a point a if it is defined at x = a and 
that is, if for every e > 0 there is a  d (e) > 0 such that  | f (x) - f (a) | < e  whenever  | x - a | < d (e).
Limits at infinity (or limits of functions as x approaches positive or negative infinity)
343.
    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. Thus,
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).
344.
   Find the vertical and the horizontal asymptote of the function
Solution:   Since,
then  x = 1  is the vertical asymptote.
And since,  
then  y = 2  is the horizontal asymptote.
345.
   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.
346.
    Find the following limits,
Solution:  The graph of the tangent function shows,
as x approaches p/2 from the left, the tangent function increases to plus infinity, while as x approaches p/2
from the right, the function decreases to minus infinity, therefore
347.
   Evaluate the limit
Solution:  
348.
   Evaluate the limit
Solution:   The graph of the arc-tangent function shows, as x tends to minus infinity the function values approach - p/2 while, as x tends to plus infinity, the function values approach p/2.
Therefore,
and
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