Sample Problems, Lesson 2.5
Sample Problems for this Lesson Course Home Page To the Notes Menu   Assignment  

2.5, #10, Given Problem.
 

2.5, #10, Given Problem.
Part (a). 

We can see that as x approaches 1 from the left that the function approaches negative infinity. 

We can also see that as x approaches 1 from the right that the function approaches  positive infinity. 
Part (b). 
As x approaches 1 from the left, the denominator will become an extremely small negative number. A positive 1 divided by an ever decreasing small negative number will yield an extremely large negative result.
As x approaches 1 from the right, the denominator will become an extremely small positive number. A positive 1 divided by an ever decreasing small posititive number will yield an extremely large positive result.
Part (c). 

We see here that our conclusions from above do seem to be correct.

2.5, # 18, Given Problem.
 

2.5, # 18, Given Problem.
=3
As x approaches infinity, the (+5) in the numerator and the (-4) will have almost no effect. The overpowering parts of the expression will be the 3x in the numerator and the x in the denominator. 3x divided by x is 3.

2.5, #32, Given Problem.
 

Find the horizontal and vertical symptotes of :
2.5, #32, Given Problem.
The only way we can get a vertical asymptote is if there is some x which causes us to be unable to calculate y. This could only happen if the denominator could be zero. So we take the time to check if and when the denominator could be zero. Using the quadratic formula, we see that no real number could cause the denominator to be zero. This means that there must not be any vertical tangents.
To find any horizontal asymptotes:

We use the technique of dividing the numerator and denominator by the largest power of x. In this case that is . (Because  the highest power in the denominator is really ). This is the same reason that when dividing both numerator and denominator by , we divided the denominator by .

Then, after simplifying, we take the limit as x approaches negative infinity and positive infinity. As x approaches infinity (positive and negative), all of the fractions with x in the denominator approach zero Therefore there are two horizontal asymptotes at .
The graph of the function confirms our work above. There really are two horizontal asymptotes.
This closer view of the above graph confirms that there really is no vertical symptote.

Here are a few of the odds:
Assignment
1-41, Odds
Lesson 2.5, Pages 139-142

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