Mathematics · 22 / 23
Limits: describe behaviour near a point
Final-year concepts · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
A limit describes behaviour as x approaches a value or infinity, even if the function is undefined at that point. Separate left and right limits; different values prevent a common two-sided limit. Dominant terms help with polynomial ratios, but indeterminate forms require transformation.
- 01Target point or infinity
- 02Domain and sides
- 03Transform if needed
- 04Conclude behaviour
WORKED EXAMPLE
From principle to reasoning
1/x tends to 0 at +∞. Near zero from above it tends to +∞; from below it tends to −∞, so no two-sided limit exists at zero.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
Find the limit of (2x+1)/x at +∞.
One hint
Divide numerator terms by x.
Reveal and understand the solution
The ratio is 2+1/x, tending to 2. y=2 is a right-hand horizontal asymptote.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Find the limit of (3x² + 1)/(x² − 2) as x → +∞.
A hint to get started
Divide top and bottom by x².
Step-by-step solution
- The quotient becomes (3 + 1/x²)/(1 − 2/x²).
- The inverse-square terms tend to zero.
- The limit is 3.
Which step did you miss? Retry without help, then explain why the method works.
Compare the left and right limits of 1/(x − 2) at 2.
A hint to get started
The denominator approaches zero with different signs.
Step-by-step solution
- From the left, x − 2 < 0: limit −∞.
- From the right, x − 2 > 0: limit +∞.
- There is no single two-sided limit; x = 2 is a vertical asymptote.
Which step did you miss? Retry without help, then explain why the method works.
Find the limit of (x² − 1)/(x − 1) at 1. Is the function defined there?
A hint to get started
Factor for x ≠ 1.
Step-by-step solution
- x² − 1 = (x − 1)(x + 1).
- For x ≠ 1 the quotient equals x + 1, with limit 2.
- The original function remains undefined at 1; limit and value differ.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
Zooming is not proof. ∞−∞ and 0/0 are indeterminate forms, not answers.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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