Read nearby behavior from tables and described graphs without confusing a limit with the value at a point.
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Try
Imagine a graph following the line y=x+2 except that its point at x=1 has been moved to height 8. Sketch the line, an open circle at (1,3), and a filled point at (1,8). What height do nearby points approach? What is the actual value at x=1? Keep these questions separate.
Predict
Predict whether moving only the filled point from height 8 to height −10 changes nearby behavior. Then consider a different graph that approaches height 3 from the left but height 6 from the right. Can one number describe both approaches?
Experiment
For x≠1, define f(x)=(x²+x−2)/(x−1), and assign f(1)=8. Compute the following values and plot them on axes of your own.
x
0.9
0.99
1
1.01
1.1
f(x)
2.9
2.99
8
3.01
3.1
The unusual middle entry is intentional. Factor the numerator as (x−1)(x+2). Away from 1, the expression equals x+2. The algebra explains the table and provides evidence for every nearby allowed input, rather than only the displayed samples.
Understand
Writing “the limit of f(x) as x approaches a is L” means that outputs become arbitrarily close to L when inputs are sufficiently close to a, excluding a itself. The function value f(a) can equal L, differ from L, or be undefined. In the example the limit is 3 and the value is 8.
The left-hand limit uses inputs smaller than a; the right-hand limit uses inputs larger than a. A two-sided finite limit exists when both one-sided limits exist and equal the same real number. For a graph with left height 3 and right height 6, the two-sided limit does not exist. Assigning a filled point at either height cannot repair this disagreement.
Consider a function equal to x+2 at the four noncentral inputs in the table and equal to 20 at all other inputs. It matches every displayed nearby sample, yet its limit at 1 is 20. A sufficiently small neighborhood avoids those four isolated sample locations. This counterexample explains why a finite table suggests behavior but cannot prove a limit without additional information about the function.
On a graph, an open circle indicates an excluded point on the described branch; a filled point records a value. A finite-resolution image can conceal jumps or oscillation. When a question gives a fully described branch rule, that rule supplies information beyond the pixels. Always identify whether your conclusion rests on a rule, a theorem, or an observation.
Apply
For each practice problem, first write three separate entries: left limit, right limit and function value. Then decide whether a common limit exists. In the table reasoning task, construct an alternative function that fits the observations. This is practice in identifying what evidence can establish, not just reading a likely pattern.
Continue
The next lesson uses limit laws and algebra to justify conclusions for whole neighborhoods. Do not substitute “the graph looks close” for an argument when a formula is available. Limits are about arbitrarily close behavior, not a particular number of decimal places.
Apply · explain · revise
Practice and reasoning
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For x≠3, f(x)=(x²−9)/(x−3), and f(3)=−4. Find the limit as x approaches 3.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A graph approaches height 2 from the left of x=1 and height 5 from the right. A filled dot is at (1,2). What is the two-sided limit?
Let g(x)=2x+1 for x<0 and g(x)=x²+4 for x≥0. Find the right-hand limit at 0.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A table gives f(0.9)=1.9, f(0.99)=1.99, f(1.01)=2.01 and f(1.1)=2.1. Explain why the table does not prove that the limit at 1 is 2.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Both branches approach (2,7), shown as an open circle; the only filled point at x=2 is (2,−1). Which pair is correct?
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
