Question

PLease help! 20 points!!
Use graphs and tables to find the limit and identify any vertical asymptotes of limit of 1 divided by the quantity x minus 4 as x approaches 4 from the left.

Answer 1

Answer: Negative infinity

note: if your teacher won't allow negative infinity, then try DNE for "does not exist"

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Explanation:

As x gets closer to x = 4 from the left side of this value, then x starts at something like x = 3 and moves to x = 3.5, then to x = 3.9, then to x = 3.99, then to x = 3.999, etc

We get closer to x = 4 but never actually get there. If you look at the table attached, then f(x) = 1/(x-4) will keep getting more negative with larger and larger negative values. This growth goes on forever without any bound. So the limit is equal to negative infinity.

As you can see on the graph below, the curve heads downward as you approach x = 4 from the left hand side. Imagine you are a point on the curve, or this point is on a rollercoaster (the curve being the track itself). As you get closer to 4 from the left side, you go downhill. There is on limit to how far downhill you can go.

note: the graph and table in the attachment below were made by the free graphing calculator program GeoGebra

Answer 2

Answer with explanation:

The given rational function is

      $y= \lim_{x \to 4^{-}} \frac{1}{x-4}$

To find the vertical Asymptotes , put

→  Denominator =0

→ x-4=0

→x=4, is the Vertical Asymptote.