Lesson 2: Finding Limits Graphically & Numerically

(a) Opening Question

Look at the function f(x) = (x² − 4)/(x − 2).

If you try to plug in x = 2, you get 0/0 — undefined. The graph has a hole at x = 2. So the function has no value at 2.

But here's the question that calculus is built on: as x gets closer and closer to 2 (from the left and from the right), what value does f(x) head toward?

Try a few inputs by hand. What is f(1.9)? What is f(2.1)? What number do these seem to be sneaking up on?

Don't worry about being at x = 2. We only care about the neighborhood around x = 2. The whole idea of a limit is this: the limit describes the behavior of f(x) near x = c, not the value of f at c. The point c itself can be a hole, a value, or anything — the limit ignores it and watches the trend. Keep this in mind as we build the tools to answer the question precisely.

(b) Core Concepts

What a limit means

The limit of f(x) as x approaches c is the single value L that f(x) gets arbitrarily close to as x gets arbitrarily close to c (but x ≠ c). We write:

lim_{x→c} f(x) = L

Read aloud: "the limit of f of x, as x approaches c, equals L."

The most important idea in this whole lesson:

A limit describes behavior NEAR c, not the value AT c.

The function might not even be defined at c (a hole), or it might be defined there but equal to something different from the limit. The limit doesn't care. It only watches what f(x) does as x closes in on c.

One-sided limits

We can approach c from two directions.

• The left-hand limit, lim_{x→c⁻} f(x), is the value f(x) approaches as x comes in from values less than c (from the left).
• The right-hand limit, lim_{x→c⁺} f(x), is the value f(x) approaches as x comes in from values greater than c (from the right).

The little superscript minus (⁻) means "from below," and plus (⁺) means "from above."

The existence rule (memorize this):

lim_{x→c} f(x) = L   if and only if   lim_{x→c⁻} f(x) = L   AND   lim_{x→c⁺} f(x) = L

The two-sided limit exists if and only if both one-sided limits exist and are equal to the same number. If the left and right limits disagree, the two-sided limit does not exist (DNE).

Reading limits off a graph

Here is a single graph with three features that show up constantly on the AP exam: a hole, a jump, and a vertical asymptote.

• Smooth curve rising from lower left, passing through a HOLE (open circle) at the point (1, 3); the curve approaches height 3 from both sides as x→1
• A solid filled dot sits at (1, 1), showing f(1) = 1 (the actual function value differs from the limit)
• To the right, the curve continues and approaches a JUMP at x = 4: from the left the curve approaches height 2 (open circle at (4, 2)); from the right the curve restarts at height 5 (solid dot at (4, 5)) and continues
• Near x = 6 there is a VERTICAL ASYMPTOTE (dashed vertical line at x = 6): the curve shoots up to +∞ as x→6 from the left and comes up from -∞ as x→6 from the right
• Label the dashed vertical line x = 6]

Read these off the graph:

• At the hole (x = 1): As x → 1 from both sides, the curve approaches height 3. So lim_{x→1} f(x) = 3. But the solid dot tells us f(1) = 1. The limit (3) and the value (1) disagree — and that's completely fine. This is the cleanest illustration of "near, not at."
• At the jump (x = 4): From the left, lim_{x→4⁻} f(x) = 2. From the right, lim_{x→4⁺} f(x) = 5. The one-sided limits disagree, so lim_{x→4} f(x) does not exist.
• At the asymptote (x = 6): From the left the curve increases without bound, so lim_{x→6⁻} f(x) = +∞. From the right it comes from below, so lim_{x→6⁺} f(x) = −∞. Because the two sides head to different unbounded behavior, lim_{x→6} f(x) does not exist.

Estimating a limit from a TABLE

When you don't have a graph, build a table that squeezes in on c from both sides. Consider the classic limit:

lim_{x→0} (sin x)/x

You cannot plug in x = 0 (you'd get 0/0). So feed in values that creep toward 0 (here x is in radians):

From both sides the outputs march toward 1. So we estimate lim_{x→0} (sin x)/x = 1. (We'll prove this exactly in Lesson 3 with the Squeeze Theorem.) Notice the dash at x = 0: the function is undefined there, but the table still pins down the limit.

A second classic — a rational function with a hole, the function from the Opening Question:

Both sides home in on 4, so lim_{x→2} (x²−4)/(x−2) = 4. (Algebraically, (x²−4)/(x−2) = x+2 for x ≠ 2, which equals 4 at x = 2 — confirming the estimate.)

When a limit DOES NOT EXIST

A two-sided limit fails to exist in three classic ways:

1. The one-sided limits disagree (a jump). Example: lim_{x→4} f(x) in the graph above.
2. Unbounded behavior (the function blows up). If f(x) → +∞ or −∞, the limit does not exist as a finite number. We still record the behavior using ±∞ (see below).
3. Oscillation. Some functions wiggle infinitely fast near c and never settle. The standard example is lim_{x→0} sin(1/x): as x → 0, 1/x races to infinity and sin(1/x) oscillates between −1 and 1 forever, never approaching a single value. The limit DNE.

Infinite limits and vertical asymptotes

Sometimes f(x) grows without bound near c. For example:

lim_{x→0} 1/x²

As x → 0 from either side, x² is a tiny positive number, so 1/x² is enormous and positive. We write lim_{x→0} 1/x² = +∞.

When a one-sided or two-sided limit is +∞ or −∞, the line x = c is a vertical asymptote of the graph. Be careful with sign on each side. For 1/x:

• lim_{x→0⁻} 1/x = −∞ (just left of 0, x is a small negative number, so 1/x is a large negative number)
• lim_{x→0⁺} 1/x = +∞ (just right of 0, x is a small positive number, so 1/x is large positive)

These disagree, so lim_{x→0} 1/x DNE. Writing +∞ or −∞ is a description of how the limit fails — it is not a number, and the limit still "does not exist" in the finite sense.

Limits at infinity and horizontal asymptotes

We can also ask what happens as x runs off toward +∞ or −∞:

lim_{x→∞} f(x)   and   lim_{x→−∞} f(x)

If f(x) → L (a finite number) as x → ±∞, then the line y = L is a horizontal asymptote. For a rational function, the end behavior is governed by the highest-degree terms. For example:

lim_{x→∞} (3x² + 2x)/(5x² − 1) = 3/5

Divide top and bottom by x²: (3 + 2/x)/(5 − 1/x²). As x → ∞, the 2/x and 1/x² terms vanish, leaving 3/5 = 0.6. So y = 0.6 is a horizontal asymptote. (We'll formalize the degree-comparison shortcut in Lesson 3.)

(c) Worked Examples

Example 1 — Reading one-sided and two-sided limits from a graph (NO CALC)

Problem. Using the graph in section (b), find each value:

(i) lim_{x→1⁻} f(x) (ii) lim_{x→1⁺} f(x) (iii) lim_{x→1} f(x) (iv) f(1) (v) lim_{x→4} f(x).

Strategy. Trace the curve toward each x-value from the left and the right. Watch the height the curve approaches, ignoring open/closed dots when finding a limit. Use the dots only for actual function values.

Solution.

• (i) Coming in from the left of x = 1, the curve rises toward height 3: lim_{x→1⁻} f(x) = 3.
• (ii) Coming in from the right of x = 1, the curve also approaches height 3: lim_{x→1⁺} f(x) = 3.
• (iii) Both one-sided limits equal 3, so lim_{x→1} f(x) = 3.
• (iv) The solid dot sits at height 1, so f(1) = 1.
• (v) From the left lim_{x→4⁻} f(x) = 2; from the right lim_{x→4⁺} f(x) = 5. They disagree.

Justification. lim_{x→4} f(x) does not exist because the left-hand limit (2) and the right-hand limit (5) are not equal. Note in part (iii)–(iv) that lim_{x→1} f(x) = 3 ≠ 1 = f(1): the limit and the function value can differ, because the limit describes behavior near x = 1, not at x = 1.

Example 2 — Estimating a limit from a table (CALC)

Problem. Use a table to estimate lim_{x→3} (x − 3)/(x² − 9).

Strategy. Direct substitution gives 0/0 (indeterminate), so build a table closing in on x = 3 from both sides.

Solution. Using the TI-84 TABLE feature (see the calculator note below):

Both sides converge to about 0.1667, i.e. 1/6.

Justification. lim_{x→3} (x − 3)/(x² − 9) ≈ 1/6 because the values approach the same number from the left and from the right. (Check: (x−3)/(x²−9) = (x−3)/[(x−3)(x+3)] = 1/(x+3), which is 1/6 at x = 3.)

Example 3 — A limit that DNE, and why (NO CALC)

Problem. Let g(x) = |x|/x. Find lim_{x→0} g(x) or explain why it does not exist.

Strategy. Simplify the piecewise behavior, then take each one-sided limit.

Solution. For x > 0, |x| = x, so g(x) = x/x = 1. For x < 0, |x| = −x, so g(x) = −x/x = −1. Therefore:

• lim_{x→0⁺} g(x) = 1
• lim_{x→0⁻} g(x) = −1

Justification. lim_{x→0} g(x) does not exist because the right-hand limit (1) and the left-hand limit (−1) are unequal. This is a jump, not an infinite limit — a reminder that "DNE" does not automatically mean ∞.

Example 4 — Limit at infinity of a rational function (NO CALC)

Problem. Find lim_{x→∞} (2x² − 5)/(x² + 3x) and state the horizontal asymptote.

Strategy. Divide numerator and denominator by the highest power of x in the denominator, here x².

Solution.

(2x² − 5)/(x² + 3x) = (2 − 5/x²)/(1 + 3/x)

As x → ∞, 5/x² → 0 and 3/x → 0, leaving (2 − 0)/(1 + 0) = 2.

Justification. lim_{x→∞} (2x² − 5)/(x² + 3x) = 2. Because the function approaches the finite value 2 as x → ∞, the line y = 2 is a horizontal asymptote of the graph. (Numerically: at x = 1000 the expression equals about 1.994, confirming the approach to 2.)

Calculator note: estimating a limit with the TI-84 TABLE

To estimate lim_{x→c} f(x) numerically:

TI-84:
1. Y= → enter the function, e.g.  Y1 = (X-3)/(X²-9)
2. 2nd → WINDOW (TBLSET):
     TblStart = 2.999   (just below c)
     ΔTbl     = 0.0005  (small step, so x values hug c)
     Indpnt: Auto,  Depend: Auto
3. 2nd → GRAPH (TABLE): read the Y1 column as X creeps toward c
4. Repeat with TblStart just ABOVE c (e.g. 3.001) to check the right side

A blank or "ERROR" in the Y1 column at exactly X = c is expected — that's the hole. The point is the trend in the rows around it, from both sides.

(d) Common Mistakes

Confusing f(c) with the limit. Students report f(c) when asked for lim_{x→c} f(x). Why it's wrong: the limit watches behavior near c; the function value at c can be a different number (or undefined). Fix: trace the curve toward c and read the height it approaches — ignore the solid dot until a question explicitly asks for f(c).

Assuming "DNE" means "∞". When a limit doesn't exist, students write ∞ automatically. Why it's wrong: a limit can DNE because of a jump (Example 3) or oscillation, with nothing going to infinity. Fix: identify which failure it is — unequal one-sided limits, unbounded behavior, or oscillation — and say so.

Sign errors on one-sided infinite limits. For lim_{x→0⁻} 1/x, students write +∞. Why it's wrong: just to the left of 0, x is a small negative number, so 1/x is a large negative number. Fix: plug in a test value like x = −0.001 to nail the sign before writing ±∞.

Forgetting radians on (sin x)/x. In Degree mode the table for (sin x)/x heads toward π/180 ≈ 0.0175, not 1. Fix: set MODE → Radian for all calculus limit work.

Reading only one side. Students check x → c from the right, see a clean value, and call it the limit. Why it's wrong: the two-sided limit exists only if both sides agree. Fix: always test both x→c⁻ and x→c⁺.

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## (e) Practice Problems

Use the graph below for problems 1–4.

• Curve enters from lower left and approaches a HOLE (open circle) at (2, 4) from both sides
• A solid dot at (2, 1) marks h(2) = 1
• The curve continues right and reaches a JUMP at x = 5: from the left it approaches height 0 (open circle at (5, 0)); from the right it restarts at height 3 (solid dot at (5, 3)) and rises
• Label tick marks clearly so heights 0, 1, 3, and 4 are readable]

(Justification) A student claims that because g(x) = (x² − 1)/(x − 1) is undefined at x = 1, the limit lim_{x→1} g(x) must not exist. Explain whether the student is correct, and find the limit if it exists. Use precise language about behavior near vs. at x = 1.

(Interpretation) The graph of a function f has a vertical asymptote at x = 3 with lim_{x→3⁻} f(x) = −∞ and lim_{x→3⁺} f(x) = +∞. A classmate writes "lim_{x→3} f(x) = ∞." Explain what is wrong with this statement and write a correct description of the behavior at x = 3.

(Justification) Build a table for f(x) = (eˣ − 1)/x approaching x = 0 from both sides. State your estimate of lim_{x→0} f(x) and justify it by referring to the left- and right-hand behavior in your table.

Let f be defined by: f(x) = x + 1 for x < 2, and f(x) = x² − 1 for x ≥ 2. Find lim_{x→2⁻} f(x), lim_{x→2⁺} f(x), and lim_{x→2} f(x) (or state DNE). Show one-sided reasoning.

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