Source: Lecture notes #21, Texas A&M University
Tags: limits, limit definition, limit existence, one-sided limits, evaluating limits, substitution, factoring, conjugate method, holes in graphs, calculus 1, CALC 157
A limit describes the y-value a function is heading towards as x approaches a particular number, regardless of whether the function actually reaches that value. A limit exists only when both the left-hand and right-hand limits agree. You can evaluate most limits through direct substitution, factoring to cancel problem terms, or multiplying by a conjugate.
Limit
The height (y-value) a function is intended to reach as x approaches a given value. Written as lim f(x) as x → c.
One-sided limit (left-hand limit)
The value a function approaches as x comes from the left (values less than c). Written as lim f(x) as x → c⁻.
One-sided limit (right-hand limit)
The value a function approaches as x comes from the right (values greater than c). Written as lim f(x) as x → c⁺.
Hole (removable discontinuity)
A point where a function is undefined or differs from the limit, but the limit itself still exists. On a graph, shown as an open circle.
DNE (Does Not Exist)
Used when a limit cannot be determined, typically because the left-hand and right-hand limits disagree.
A limit is the y-value a function intends to reach as x approaches a specific number.
It does not matter whether the function actually arrives at that value. What matters is the trend.
Example with f(x) = x²:
Different x-values produce different heights.
When x = 2, y = 4, so lim f(x) as x → 2 = 4.
Example with a hole:
Consider (x² – 6x + 8) / (x – 2), which factors to (x – 4)(x – 2) / (x – 2).
At x = 2, the function is undefined (division by zero), creating a hole at (2, –2).
The limit still exists at x = 2 because the function is heading towards –2 from both sides.
lim g(x) as x → 2 = –2, even though g(2) is undefined.
A limit exists when you approach a single point from both sides of the x/y axes and get the same value.
A limit does not exist when the function "breaks," i.e. the left-hand and right-hand limits disagree.
How to check:
Evaluate the left-hand limit (approaching from below).
Evaluate the right-hand limit (approaching from above).
If they match, the general limit equals that shared value.
If they differ, the limit DNE.
Example of DNE:
If lim f(x) as x → 4⁺ = 2 (from the right) and lim f(x) as x → 4⁻ = 1 (from the left), the values do not match, so lim f(x) as x → 4 = DNE.
Key rule: A limit can still exist when there is a hole on the graph. Holes affect function value, not limit value.
Three core techniques, in order of complexity:
Technique 1 – Direct substitution
Plug the target x-value directly into the function.
If you get a real number (no division by zero, no 0/0), that is the limit.
Example: lim (x² – 1) / (2x + 3) as x → 0 = (0 – 1) / (0 + 3) = –1/3.
Technique 2 – Factoring
Use this when substitution gives 0/0 (an indeterminate form).
Factor the numerator and denominator, cancel the common factor, then substitute.
Example: lim (x² – 6x + 8) / (x – 2) as x → 2
Factor: (x – 4)(x – 2) / (x – 2)
Cancel (x – 2): leaves (x – 4)
Substitute x = 2: 2 – 4 = –2
lim = –2
Technique 3 – The conjugate method
Use this when the expression contains a square root and substitution gives 0/0.
Multiply the numerator and denominator by the conjugate of the radical expression.
Example: lim (√x – 4) / (x – 16) as x → 16
Multiply top and bottom by (√x + 4)
Numerator becomes: x – 16 (difference of squares)
Simplifies to: 1 / (√x + 4)
Substitute x = 16: 1 / (√16 + 4) = 1 / (4 + 4) = 1/8
Limit notation:
lim f(x) as x → c = L
Factoring pattern (difference of squares / quadratic):
x² – 6x + 8 = (x – 4)(x – 2)
Conjugate identity:
(√a – b)(√a + b) = a – b²
⚠️ A limit can exist even when the function is undefined at that point. Holes do not kill limits.
⚠️ Always try substitution first. Factoring and the conjugate method are for when substitution returns 0/0.
⚠️ "DNE" only applies when left-hand and right-hand limits disagree, or when the function heads to infinity (covered in Lesson 4).
⚠️ When using the conjugate method, make sure you multiply both the numerator and the denominator, not just one.
⚠️ After factoring and cancelling, you must still substitute the x-value to get the final answer.
Q: What is a limit in plain terms?
A: The y-value a function is heading towards as x gets closer to a particular number, whether or not it actually reaches that value.
Q: If f(2) is undefined, can lim f(x) as x → 2 still exist?
A: Yes. A limit depends on the values the function approaches near x = 2, not the value at x = 2 itself. A hole does not prevent the limit from existing.
Q: When does a limit not exist?
A: When the left-hand limit and the right-hand limit are different values (they do not agree), or when the function increases or decreases without bound.
Q: Evaluate lim (x² – 6x + 8) / (x – 2) as x → 2.
A: Factor to (x – 4)(x – 2) / (x – 2), cancel (x – 2), substitute x = 2 into (x – 4) to get –2.
Q: When should you use the conjugate method?
A: When the expression involves a square root and direct substitution gives the indeterminate form 0/0. Multiply by the conjugate to eliminate the radical.
Q: Evaluate lim (√x – 4) / (x – 16) as x → 16.
A: Multiply by (√x + 4)/(√x + 4). Numerator becomes x – 16, which cancels with the denominator. Left with 1/(√x + 4). Substitute x = 16 to get 1/8.
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