1.3 Estimating Limit Values from Graphs

Learning Objectives

  • Estimate a limit by looking at a function’s graph.
  • Recognize holes (removable discontinuities) and jumps (non-existent limits).
  • Distinguish between one-sided and two-sided limits visually.

Motivation

Before calculating limits algebraically, we often estimate limits from graphs. This helps build intuition about what a limit really means — it’s the value the graph is approaching, not necessarily the value of the function at that point.

How to Estimate Limits from Graphs

  1. Trace the curve with your eyes as \(x\) approaches the value of interest from the left and from the right.
  2. If the \(y\)-values approach the same number, that’s the limit.
  3. If they approach different numbers → the limit does not exist (jump discontinuity).
  4. If there’s a hole but both sides approach the same height → the limit exists, even if the point is missing.

Example 1: A Hole

For \( f(x) = \tfrac{x^2-1}{x-1} \), the graph looks like a line \(y=x+1\) with a hole at \(x=1\).

Visually, both sides get close to \(2\). So \( \lim_{x\to 1} f(x) = 2 \).

Example 2: A Jump

For the step function \( g(x) = \begin{cases}2, & x<1 \\ 4,& x>1\end{cases} \), the left side approaches 2, the right side approaches 4. Since they differ, \( \lim_{x\to 1} g(x) \) does not exist.

Interactive Graph: Drag \(a\) and Estimate

Use the slider to move the point \(a\). Watch the left (\(x=a-h\)) and right (\(x=a+h\)) approach points and estimate the limit from the graph.

1.00
0.60

Move \(a\) near different discontinuities. Does the limit exist? Estimate it visually.

Quick Reference

  • If the graph has a hole, the limit still exists (at the hole’s \(y\)-value).
  • If the graph has a jump, the left/right limits differ → the limit DNE.
  • If the graph goes to \(\infty\) or \(-\infty\), the limit is infinite (diverges).

Try It

  1. Look at a graph of \( y = |x|/x \). What is \( \lim_{x\to 0^-} y \) and \( \lim_{x\to 0^+} y \)? Does the two-sided limit exist?
  2. Estimate \( \lim_{x\to 2} f(x) \) if the graph shows a hole at (2,3).
  3. On a graph where the curve rises without bound near \(x=1\), describe \( \lim_{x\to 1} f(x) \).

Key Takeaways

  • Limits describe the approach, not necessarily the actual function value.
  • Use the graph to compare left and right sides: same → limit exists; different → limit DNE.
  • Holes do not break a limit; jumps do.