Estimating Limit Values from Graphs
Introduced| English | Meaning |
|---|---|
| left-hand limit/left hænd ˈlɪmɪt/ | The value a function approaches as its input tends to a point through smaller values. |
| right-hand limit/raɪt hænd ˈlɪmɪt/ | The value a function approaches as its input tends to a point through larger values. |
| hole/həʊl/ | A removable gap in a graph where a finite limit exists but the function is undefined or assigned a different value. |
| jump/dʒʌmp/ | A discontinuity where finite left-hand and right-hand limits exist but are different. |
| unbounded/ʌnˈbaʊndɪd/ | Not restricted by a finite upper bound, lower bound or both, as specified by the context. |
| oscillation/ˌɒsɪˈleɪʃn/ | Repeated variation between values, sometimes without approaching one limiting value. |
Read the limit straight off the picture
- Given a graph, you can often see a limit without any algebra.
- Put your finger on the curve a little left of $x=c$ and trace it toward $c$: what height are you heading for?
- Do the same from the right. If both fingers aim at the same height, that height is the limit.
- The trick: watch where the curve is going, and ignore any single plotted dot at $x=c$.
Trace the curve toward a point
y = ax² + bx + c
Slide toward an input from the left and the right — the limit is the shared height the curve approaches, whatever a lone dot might say.
To read $\lim_{x\to c} f(x)$ from a graph, you should look at...
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A limit is the shared height approached from both sides — the dot at $c$ is irrelevant.
Left and right, read separately
- Trace from the left only → the left-hand limit 左极限 $\displaystyle\lim_{x\to c^-}f(x)$.
- Trace from the right only → the right-hand limit 右极限 $\displaystyle\lim_{x\to c^+}f(x)$.
- On a graph a filled dot ● means the point is on the curve; an open dot ○ (a hole 空洞) means it is not.
- The limit ignores whether the dot is filled or open — it only cares about the approach.
A graph rises steadily and, just left of $x=5$, the curve is passing through heights $2.9, 2.99, 2.999$. Estimate $\lim_{x\to 5^-} f(x)$.
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The left-hand values close in on $3$, so the left-hand limit is $3$.
When the limit does not exist
- Three classic graph shapes make a limit fail (DNE):
- A jump 跳跃: the two sides trace to different heights.
- An unbounded 无界 blow-up: the curve rockets to $\pm\infty$ near a vertical asymptote.
- An oscillation 振荡: the curve wiggles infinitely fast (like $\sin\frac1x$) and never settles.
If the left side of a graph heads to $2$ and the right side heads to $6$, the limit does not exist because of a ____.
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Different one-sided heights means a jump discontinuity, so the two-sided limit is DNE.
From a graph, select all features that make $\lim_{x\to c} f(x)$ fail to exist.
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Blow-ups, jumps, and oscillation all kill a limit. A removable hole does not — both sides still agree.
The dot at $c$ is a decoy
- The height the curve reaches for, $\lim_{x\to c}f(x)$, is a separate question from the plotted value $f(c)$.
- A curve can head smoothly toward $4$ while a lone dot sits at $f(c)=1$ — the limit is still $4$.
- $f(c)$ can even be undefined (a hole) while the limit exists.
- So always answer two questions separately: where is the curve going? and is there a dot, and where?
If the graph shows an open hole at $(2,4)$ and a filled dot at $(2,1)$, then $\lim_{x\to 2} f(x) = 4$.
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The curve approaches height $4$ from both sides, so the limit is $4$; the filled dot is only $f(2)=1$.
A smooth curve passes through height $7$ at $x=3$ with no hole or dot elsewhere. What are $f(3)$ and $\lim_{x\to 3} f(x)$?
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With no hole or jump, the curve is continuous there, so limit $= f(3) = 7$.
A common trap: reading $f(c)$ (the plotted dot) when the question asks for $\lim_{x\to c}f(x)$ (the approach). They are different. If you see a filled dot floating above a smooth curve, the limit is the curve's height — not the dot's.
From a graph: near $x=2$ the curve rises to a hole at height $4$, and a filled dot is plotted at $(2,\,1)$.
- Left-hand limit: heading to $4$. Right-hand limit: heading to $4$.
- Both agree, so $\displaystyle\lim_{x\to 2}f(x) = 4$.
- But $f(2) = 1$ (the filled dot). Limit $\neq$ function value here.
To read a limit off a graph, trace the curve toward $c$ from both sides and ask what height you head for; the plotted dot at $c$ is a decoy. The limit is DNE when the sides give different heights (jump), the curve is unbounded, or it oscillates endlessly.
Write from a jump
- The left side of a graph heads toward 2.
- The right side heads toward 6.
Does the two-sided limit exist, and why?
Disagreeing one-sided limits mean no two-sided limit.
Left height 2 and right height 6 means the two-sided limit is what?
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The sides must agree. Other options — 4 (the average); 2 (the left height).