Topic 1.3 Estimating Limit Values from Graphs: use a graph to estimate one-sided and two-sided limits, including cases where the function value differs from the limit or does not exist.

What this topic is asking

The College Board (Topic 1.3) wants you to read limits off a graph. You should find one-sided and two-sided limits by tracing the curve toward a point, recognize that the limit is the height the curve approaches (not the plotted dot), and identify when a limit does not exist because of a jump or an asymptote.

How to read a limit off a graph

Put your pencil on the curve away from $x = a$ and slide it toward $a$. The $y$-value your pencil heads toward is the one-sided limit on that side:

• Slide in from the left to get $\lim_{x \to a^-} f(x)$.
• Slide in from the right to get $\lim_{x \to a^+} f(x)$.

If both slides aim at the same height $L$, then $\lim_{x \to a} f(x) = L$. If they aim at different heights, the two-sided limit does not exist.

Open versus filled circles

This is the single most tested idea in Topic 1.3: the limit reads the trend, the filled dot reads the value, and they need not agree.

Cases where the limit does not exist

• Jump: the left and right pieces approach different heights (a step). The two-sided limit DNE.
• Vertical asymptote: the curve shoots up to $+\infty$ or down to $-\infty$ near $a$. The finite limit DNE (you may still describe it as $\pm\infty$).
• Oscillation: the curve wiggles infinitely fast near $a$ and never settles on a height.

Limits versus values across a whole graph

A common exam graph packs several features into one picture and asks a battery of questions: the limit here, the value there, the one-sided limits at a jump. The discipline that keeps you accurate is to answer each part with the right object. For "$\lim_{x \to a} f(x)$" you trace the curve in from both sides and report the approaching height. For "$f(a)$" you find the single plotted point - the filled circle - and report its height, or "undefined" if no filled point sits at $x = a$. For a one-sided limit you trace from only the named side. Because a single point on the graph can be an open circle (a height approached) sitting above or below a separate filled circle (the actual value), keeping "approach" and "value" in separate mental columns is what prevents the classic mix-ups. Work left to right through the requested parts, naming which object each one wants before you read the graph.

Reporting your reading

When an AP question asks you to estimate from a graph, give the height as a number and, if asked, state existence with a justification that compares the two sides. Be explicit: "the left-hand limit is $2$ and the right-hand limit is $6$, so the two-sided limit does not exist." For an asymptote, write "$\lim_{x \to a^+} f(x) = +\infty$, so the limit does not exist." Graph-reading questions reward precise language, so name the side you used and compare the two sides whenever existence is in question; a vague "the limit is about $5$" without acknowledging a jump can cost the justification point even when the number is defensible.