Limits and Derivatives · Lesson 2 of 8
Understanding Limits from Both Sides
“Read tables and graphs to decide what a function approaches, even when its value at the point is different.”
• Read and explain limit notation. • Determine left-hand and right-hand limits from tables, graphs, and piecewise formulas. • Decide whether a two-sided limit exists. • Distinguish a limit from the function value at the point. • Respect the domain when choosing a direction of approach.
Nearby values answer the limit question
A limit asks what the output approaches as the input moves close to a specified number. It does not require us to use the output at that number. This distinction is essential: a function can have a perfectly clear limiting value at a point where it is not defined.
For example, when x is close to zero, x² is close to zero. When x moves closer still, x² moves closer still. We describe this by saying that the limit of x² as x tends to zero is zero. The arrow records an approach, rather than an instruction to set x equal to the target immediately.
A value approached by the outputs f(x) as allowed inputs x get arbitrarily close to a, with x different from a. For an interior point, both sides must approach the same value.
If g(x)=|x| for x≠0, g(0) is absent from its domain. Nevertheless the outputs on both sides of zero approach zero. The missing value at zero does not prevent the limit from being zero.
Similarly, h(x)=(x²−4)/(x−2) is undefined at x=2. For x≠2 it is exactly x+2, because x²−4=(x−2)(x+2). Its outputs near 2 approach 4. The graph is a straight line with one missing point, not a graph that fails to approach a height.
Left and right are separate tests
To approach a number from the left, use inputs smaller than it. To approach from the right, use larger inputs. These can lead to different outputs if a function uses different formulas on the two sides. The value at the target itself cannot make unequal side limits agree.
Problem
Let f(x)=1 for x≤0 and f(x)=2 for x>0. Does the limit at zero exist?
- 1.For x<0, every output is 1, so the left-hand limit is 1.
- 2.For x>0, every output is 2, so the right-hand limit is 2.
- 3.These values are different, so the two-sided limit does not exist. The assigned value f(0)=1 does not change this conclusion.
A limit can fail to exist even when f(a) is defined. It can also exist when f(a) is undefined. Ask about nearby outputs before asking about the value at the target.
Patterns in numerical tables
Tables can suggest a limit by displaying inputs progressively closer to a. They are particularly useful when we are learning the meaning of the notation. A finite table alone is evidence rather than a universal proof: the formula or graph must support the pattern beyond the sampled entries.
| x near 5 | 4.9 | 4.95 | 4.99 | 4.995 | 5.001 | 5.01 | 5.1 |
|---|---|---|---|---|---|---|---|
| x+10 | 14.9 | 14.95 | 14.99 | 14.995 | 15.001 | 15.01 | 15.1 |
For x+10 near x=5, the left outputs approach 15 from below and the right outputs approach 15 from above. Both side limits are 15. The value f(5)=15 also happens to agree with the limit.
| x near 1 | 0.9 | 0.99 | 0.999 | 1.001 | 1.01 | 1.1 |
|---|---|---|---|---|---|---|
| x³ | 0.729 | 0.970299 | 0.997002999 | 1.003003001 | 1.030301 | 1.331 |
For x³ near x=1, the common limiting output is 1. Cubing the nearby inputs preserves the approach to that value. In both of these examples the function value and the limit agree, but that agreement must not be mistaken for the definition of a limit.
| x near 2 | 1.9 | 1.95 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 |
|---|---|---|---|---|---|---|---|
| 3x | 5.7 | 5.85 | 5.97 | 5.997 | 6.003 | 6.03 | 6.3 |
For the constant function f(x)=3, every nearby value is already 3, so its limit at any real input is 3. A limit does not require a visible change in the outputs; it describes their approach, including the case where they remain fixed.
| x near 1 | 0.9 | 0.99 | 0.999 | 1.01 | 1.1 | 1.2 |
|---|---|---|---|---|---|---|
| x²+x | 1.71 | 1.9701 | 1.997001 | 2.0301 | 2.31 | 2.64 |
The limit of x²+x at 1 is 2. Separately, the limits of x² and x at 1 are each 1, so the limit of their sum agrees with the sum of their limits. Writing x²+x as x(x+1) also suggests that a product behaves similarly. These observations motivate the limit laws that will be justified and used in the next lesson.
| x near π/2 | π/2−0.1 | π/2−0.01 | π/2+0.01 | π/2+0.1 |
|---|---|---|---|---|
| sin x (radians) | 0.9950 | 0.99995 | 0.99995 | 0.9950 |
The sine outputs approach 1 near π/2. The inputs are angles measured in radians. In the next table we again use radians for the cosine term; adding x to cos x gives outputs approaching 1 near zero.
| x near 0 | −0.1 | −0.01 | −0.001 | 0.001 | 0.01 | 0.1 |
|---|---|---|---|---|---|---|
| x+cos x (rounded) | 0.895004 | 0.989950 | 0.999000 | 1.001000 | 1.009950 | 1.095004 |
The limit of x is zero and the limit of cos x is 1 as x tends to zero. The table supports a limit of 0+1=1 for their sum. Numerical entries have been calculated directly from the formula so that rounding does not hide the actual trend.
A function value need not match the limit
Two important cases complete the picture. A jump can give different side limits even if one value is assigned at the join. A hole can have matching side limits even if a completely different value is assigned at that point.
Problem
Let f(x)=x−2 for x<0, f(0)=0, and f(x)=x+2 for x>0. Find the limit at zero.
- 1.On the left, x−2 approaches −2. On the right, x+2 approaches 2.
- 2.The branches approach different heights, so the two-sided limit does not exist.
- 3.Although f(0)=0, this isolated value says nothing about whether the two nearby branches meet.
| x | −0.1 | −0.01 | −0.001 | 0.001 | 0.01 | 0.1 |
|---|---|---|---|---|---|---|
| f(x) | −2.1 | −2.01 | −2.001 | 2.001 | 2.01 | 2.1 |
Problem
Let f(x)=x+2 for x≠1 and f(1)=0. Find the limit at 1 and compare it with f(1).
- 1.Every input close to 1 but different from 1 uses x+2.
- 2.As x approaches 1 from either side, x+2 approaches 3.
- 3.Thus the limit is 3 while the function value is 0. A single altered point does not alter the nearby approach.
| x near 1 | 0.9 | 0.99 | 0.999 | 1.001 | 1.01 | 1.1 |
|---|---|---|---|---|---|---|
| f(x) | 2.9 | 2.99 | 2.999 | 3.001 | 3.01 | 3.1 |
Let the domain guide the approach
For f(x)=1/x² with domain x>0, only positive inputs are allowed. There is no left-side data near zero in this domain. As these positive inputs approach zero, the outputs grow without any finite bound. This is written as a limit of +∞, which describes unbounded growth rather than an ordinary real-number value.
This is a qualitative observation here, not a new set of infinite-limit techniques. It also reminds us to check the domain before asking for a side limit: one must approach through inputs for which the function exists.
| Positive input x | 1 | 0.1 | 0.01 | 10⁻ⁿ |
|---|---|---|---|---|
| 1/x² | 1 | 100 | 10000 | 10²ⁿ |
Problem
Let f(x)=a+bx for x<1, f(1)=4, and f(x)=b−ax for x>1. Find a and b so that the limit at 1 equals f(1).
- 1.The left formula approaches a+b, so require a+b=4.
- 2.The right formula approaches b−a, so require b−a=4.
- 3.Subtract the equations to obtain 2a=0. Then a=0 and b=4. Both side limits and the point value are now 4.
Quiz
A function is undefined at a, but both side limits are 7. What follows?
If the left limit is −2 and the right limit is 2, what is the two-sided limit?
What does x→a⁻ mean?
For f(x)=x+2 except f(1)=0, what is the limit at 1?
What is the limit of the constant function 3 at x=2?
Which conclusion is justified by a finite table alone?
Practice Problems
- For f(x)=2x+3 when x≤0 and f(x)=3(x+1) when x>0, find the limits at 0 and 1.
- For f(x)=x²-1 when x≤1 and f(x)=-x²-1 when x>1, determine whether the limit at 1 exists.
- Let f(x)=|x|/x for non-zero x and f(0)=0. Find each one-sided limit at zero and decide whether the two-sided limit exists.
- Repeat the preceding investigation for f(x)=x/|x| when x is non-zero, with f(0)=0. Explain the agreement between the answers.
- Find lim[x→5] (|x|-5). Explain which formula for |x| is relevant close to 5.
- For f(x)=a+bx on x<1, f(1)=4, and f(x)=b−ax on x>1, find the constants making the limit equal to f(1), and verify both sides.
- Let f(x)=x+1 for x<0, f(0)=0, and f(x)=x−1 for x>0. At which real numbers a does the limit as x→a exist?
- For f(x)=mx²+n on x<0, f(x)=nx+m on 0≤x≤1, and f(x)=nx³+m on x>1, find all integer pairs (m,n) for which the limits at 0 and 1 both exist.
Key Takeaways
• A limit is controlled by nearby values, not by a single value at the target. • At an interior point, the two-sided limit exists only when both side limits exist and agree. • A missing point, a different point value, and a jump are distinct situations. • Numerical tables suggest patterns; formulas and graphs explain them. • Only inputs inside the domain can participate in an approach.