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Functions, Graphs, and Limits

Functions, Graphs, and Limits

At a Glance - Vertical Asymptotes vs. Holes

Both vertical asymptotes and holes are places that the curve can't quite seem to touch. Holes occur at places where the limit of the function exists, but the function itself does not. For rational functions, holes correspond to the roots (or zeros) of the denominator that cancel out entirely during simplification. Vertical asymptotes occur at places where the limit of the function is ∞ or -∞, which happen at the roots of the denominator that are left over after simplification. It's an important distinction.

Example 1

Find all vertical asymptotes and/or holes of the function 


Example 2

Find all vertical asymptotes and/or holes of the function 


Exercise 1

Find all vertical asymptotes and holes for the rational function below. Give both the x and y coordinates for the holes.

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Exercise 2

Find all vertical asymptotes and holes for the rational function below. Give both the x and y coordinates for the holes.


Exercise 3

Find all vertical asymptotes and holes for the rational function below. Give both the x and y coordinates for the holes. 


Exercise 4

Find all vertical asymptotes and holes for the rational function below. Give both the x and y coordinates for the holes.


Exercise 5

Find all vertical asymptotes and holes for the rational function below. Give both the x and y coordinates for the holes.


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