Calculus I · Limits and Continuity · review
Common Limits Exam Errors
Common Exam Errors and Their Repairs
| Error | Why it fails | Repair |
|---|---|---|
| Using the filled point as the limit | The point gives , not nearby behavior | Trace the graph from both sides first |
| Calling the answer | It is an indeterminate form | Simplify, then substitute again |
| Cancelling terms across addition | Cancellation applies to factors | Factor the entire numerator or denominator |
| Ignoring left and right sides | Two-sided limits require agreement | Compute both one-sided limits |
| Writing at negative infinity | The principal square root is nonnegative | Use for |
| Using degrees in | The fundamental limit is a radian statement | Convert or use radian measure |
| Applying IVT without continuity | Jumps can skip intermediate values | State continuity on the entire closed interval |
| Claiming IVT gives the exact root | It gives existence only | Use bisection or another numerical method for location |
| Using l'Hospital's Rule too early | It may be unavailable, circular, or hide the needed algebra | Use first-unit methods requested by the course |
After the explanation
Use the section idea
Can you diagnose the limit type and justify a method before beginning the algebra?
A mixed problem is a classification task before it is a calculation: direction, substitution result, structure, and required conclusion determine the route.
Name the limit type and first legal move in a margin note, then solve and check whether the conclusion matches the graph or sign behavior.
Pattern matching without diagnosis makes similar-looking problems blur together and hides whether the error was conceptual, algebraic, or strategic.
You are exam-ready when you can choose a method without a section label, explain the choice, and correct a miss by naming its exact cause.
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