Finding Limits from Graphs

Finding a limit from a graph means looking at what height (y-value) a function is approaching as \(x\) gets closer and closer to some target value \(a\), without ever needing to plug numbers into a formula. Instead of computing, you trace the curve with your eye or a pencil, moving in from both the left side and the right side of \(x = a\), and watch where the curve is heading. If you have not yet seen the formal idea behind this, it helps to review the introduction to limits before working through graphs.

This graphical approach is powerful because a picture instantly shows things that are easy to miss in an equation: sudden jumps, holes in the curve, or the graph shooting off toward infinity. All of these show up clearly once you know what to look for.

Every limit question really has two halves. The left-hand limit, written \(\lim_{x \to a^-} f(x)\), asks what height the graph approaches as \(x\) slides in toward \(a\) from values smaller than \(a\). The right-hand limit, written \(\lim_{x \to a^+} f(x)\), asks the same question approaching from values larger than \(a\).

The two-sided limit, \(\lim_{x \to a} f(x)\), exists only when both one-sided limits exist and are equal:

\(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \quad \Rightarrow \quad \lim_{x \to a} f(x) = L\)

If the two sides approach different heights, the two-sided limit simply does not exist at that point, even though the function might still be perfectly defined there.

The easiest case is a smooth, unbroken curve. As you trace along the graph below toward \(x = 2\) from either direction, both sides settle on the same height, so the limit exists and equals that height.

Because the curve has no break, hole, or jump near \(x = 2\), the limit as \(x \to 2\) matches the actual function value there. This is exactly the situation where the direct substitution property works algebraically, since the graph is continuous at that point.

Piecewise functions are the classic place where graphical limits get interesting, because the pieces do not always connect. In the graph below, the function follows one line for \(x < 2\) and a different line for \(x \ge 2\), leaving a jump at \(x = 2\).

Reading this graph: as \(x \to 2^-\) (blue line), the curve approaches a height of 3, shown by the open circle, which means that value is not actually reached from that side. As \(x \to 2^+\) (red line), the curve approaches a height of 1, and the closed dot shows that \(f(2) = 1\) is the true function value. Since the two sides disagree,

\(\lim_{x \to 2^-} f(x) = 3, \qquad \lim_{x \to 2^+} f(x) = 1, \qquad \lim_{x \to 2} f(x) \) does not exist.\(\)

Notice that \(f(2) = 1\) exists as a function value, but that has no bearing on whether the limit exists. This distinction between a function's value at a point and its limit near that point is central to understanding continuity, where a function is continuous only when the two agree.

When you scan a graph for a limit, three visual features tell you almost everything:

• Open circle (hole): the graph approaches that height but the function is undefined or defined differently there. The limit can still exist, matching the height of the open circle.
• Jump: the graph approaches two different heights from the left and right, as in Example 2 above. The two-sided limit does not exist at the jump.
• Vertical asymptote: the graph shoots up or down without bound as \(x\) approaches a value. These are covered in detail in infinite limits and vertical asymptotes, since the limit is described as positive or negative infinity rather than a finite number.

It is also worth remembering that limits describe local behavior near a point, not behavior far away. For graphs that flatten out toward a horizontal line as \(x\) grows without bound, that is a separate idea handled on the limits at infinity page.

1. Locate the target \(x\)-value on the horizontal axis.
2. Trace the curve as \(x\) approaches that value from the left; note the height it approaches.
3. Trace the curve as \(x\) approaches that value from the right; note the height it approaches.
4. Compare the two heights. If they match, that shared height is the limit. If not, the limit does not exist.
5. Ignore whether that exact point is filled in, open, or missing altogether; the limit only cares about the approach, not the landing.

With practice, this scanning process becomes automatic, and reading limits straight off a graph becomes one of the fastest tools in calculus for checking whether a function behaves nicely at a point.

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