Notes on Limits: One-Sided, Two-Sided, and Infinite Limits

• Limits describe the value that f(x) gets arbitrarily close to as x approaches a, not necessarily the value f(a) itself.
• Infinity is not a number; it is a way to describe unbounded growth of a function.

Directional vs. Nondirectional (Two-Sided) Limits

• Left-hand limit at a: $\lim_{x\to a^-} f(x)$
• Right-hand limit at a: $\lim_{x\to a^+} f(x)$
• If both exist and are equal to the same value L, then the two-sided limit exists and is $\lim_{x\to a} f(x) = L$.
• If one-sided limits exist but are different, the two-sided limit does not exist.
• The function value at the point does not determine the limit; the limit concerns values arbitrarily close to the point from each side.

Infinite Limits: When the Limit Does Not Approach a Fixed Number

• The function grows without bound as we approach a from the left or from the right, i.e., the limit is infinite.
• The limit may be:
  • $\lim_{x\to a^+} f(x) = +\infty$ or $-\infty$,
  • $\lim_{x\to a^-} f(x) = +\infty$ or $-\infty$.
• If both one-sided limits are the same type of infinity (both +\infty or both -\infty), the two-sided limit is written as $\lim_{x\to a} f(x) = +\infty\quad\text{or}\quad -\infty$.
• If one side tends to +\infty and the other to -\infty (or if signs differ), the two-sided limit does not exist.

How to Determine the Sign of an Infinite Limit

• To analyze the limit, factor or shift to reveal the sign of the small quantity in the denominator. Examine the sign of the expression that goes to zero in the denominator from the appropriate side.
  • Example: For $f(x) = \frac{1}{x-3}$
  • As x → 3^+, x-3 > 0, so $\lim_{x\to 3^+} \frac{1}{x-3} = +\infty.$
  • As x → 3^-, x-3 < 0, so $\lim_{x o 3^-} \frac{1}{x-3} = -\infty.$

Exam Conventions for Infinite Limits

• If both left and right limits are +\infty (or both -\infty): the two-sided limit is written as $\lim_{x\to a} f(x) = +\infty$ (or $-\infty$) and is considered to exist in the sense of an infinite limit.
• Be precise: use $+\infty$ or $-\infty$ with the appropriate side (e.g., $\lim_{x\to a^+} f(x) = +\infty$) and only declare a two-sided limit as +\infty or -\infty when both sides agree.

Key Takeaways

• Limits describe values approached, not necessarily the function value at the point of interest.
• Left-hand and right-hand limits can exist separately even if the two-sided limit does not.
• If both left and right limits exist and are equal, the two-sided limit exists and equals that common value.
• If left and right limits diverge to infinity (or negative infinity) and match in type on both sides, the two-sided limit is an infinite limit; otherwise the two-sided limit does not exist.
• Infinity is not a number; it is a symbol indicating unbounded growth, and you should use +\infty or -\infty to describe the direction of growth.

Quick Reference Notation

• Left limit: $\lim_{x\to a^-} f(x)$
• Right limit: $\lim_{x\to a^+} f(x)$
• Two-sided limit: $\lim_{x\to a} f(x)$ (exists if both one-sided limits exist and are equal)
• Infinite limits: $\lim<em><em>{x\to a^+} f(x) = \pm\infty\qquad\text{and/or}\qquad \lim</em></em>{x\to a^-} f(x) = \pm\infty$
• Note: Infinity is not a number; do not perform arithmetic with it.