How do you find vertical asymptotes using limits?

Answer 1
The vertical line #x=a# is a vertical asymptote of $f(x)$ if either #lim_{x to a^-}f(x)=pm infty# or #lim_{x to a^+}f(x)=pm infty#.
So, we need to find #a#-values such that either the left-hand limit or the right-hand limit is #pm infty#.
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Answer 2

To find vertical asymptotes using limits, follow these steps:

  1. Determine the rational function's expression.
  2. Set the denominator equal to zero and solve for the variable(s).
  3. The values obtained in step 2 are potential vertical asymptotes.
  4. Take the limit of the function as it approaches each potential asymptote from both sides.
  5. If the limit approaches positive or negative infinity, then there is a vertical asymptote at that value.
  6. Repeat steps 4 and 5 for each potential asymptote to confirm their presence.

Note: It is important to consider any restrictions on the domain of the function, as these may affect the existence of vertical asymptotes.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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