Limits practice question 5
Evaluate the limit.
Two things decide a limit like this: whether the size blows up, and the sign on each side of . Check both for the top and the squared bottom.
Substituting gives . What sign does have just left of , and just right of it?
Theory UBC Math 100 §2.1CLP-1 §1.3Key results
Worked solution
Strategy
The numerator tends to −4 and the denominator to 0, so the size grows without bound. A squared denominator is positive on both sides, so the two one-sided limits agree.
Working
A square is never negative, so the denominator is small and positive on both sides.
Negative over small positive is large and negative, from either side.
The one-sided limits agree, so the two-sided limit is −∞, and x = −3 is a vertical asymptote.
Answer
The answer is negative infinity.
Check
At x = −3.01 the quotient is −40,100 and at x = −2.99 it is −39,900: negative and huge on both sides.
Where this goes wrong
Writing “does not exist” because the values blow up. When both sides head to −∞ the limit is −∞; “does not exist” is for limits that fail in other ways, such as one-sided limits that disagree.
The idea behind it
The limits lesson, a figure to move, in a few steps
Leans on
Sign analysis near a root, from Functions, graphs and inverses
Try a similar question
Evaluate the one-sided limit.Saved in this browser only.