CHANGE AND LIMITS 04
Infinite Limits
An infinite limit describes function values that grow without bound as the input approaches a finite value.
The main idea
For example,
\[\boxed{\lim_{x\to0}\frac1{x^2}=+\infty}.\]This does not mean infinity is an ordinary function value. It describes unbounded behaviour near \(x=0\).
1. Growing without bound
As \(x\) gets closer to zero, \(1/x^2\) becomes larger: \(x=0.1\) gives \(100\), while \(x=0.01\) gives \(10000\).
2. Vertical asymptotes
If a function becomes unbounded as \(x\to a\), the line \(x=a\) is a vertical asymptote. For \(1/x^2\), the vertical asymptote is \(x=0\).
3. Positive and negative infinity
For \(1/x\),
\[\lim_{x\to0^+}\frac1x=+\infty,\qquad\lim_{x\to0^-}\frac1x=-\infty.\]The two sides behave differently, so the two-sided limit is not a single infinite limit.
4. Example away from zero
For
\[f(x)=\frac1{(x-2)^2},\]the denominator approaches zero as \(x\to2\), while remaining positive. Hence
\[\lim_{x\to2}\frac1{(x-2)^2}=+\infty.\]5. Recognising the behaviour algebraically
When a denominator approaches zero, inspect its sign from each side. A very small positive denominator produces a large positive quotient; a very small negative denominator can produce a large negative quotient.
→ quotient may → \(+\infty\)
denominator → \(0^-\)
→ sign must be checked
Infinite limits describe unbounded behaviour near a finite input. One-sided behaviour and signs matter, and vertical asymptotes are the graphical signature of this type of limit.