Calculus
Confusing a Function's Limit with Its Value
Students replace a limit with the function value at that point, even when the graph has a hole or jump there.
Quick check
Why students make it
It feels natural to plug in the input because many limits can be found by direct substitution. The mistake is assuming the value at the point controls the behavior near the point.
The rule done right
A limit depends on what approaches as gets close to , not necessarily on . The value can be different, undefined, or the same as .
Related resources
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Calculus, Limits Numerically and Graphically Infographic -
Calculus, One-Sided Limits Infographic -
Calculus, Continuity at a Point Infographic -
Calculus, Types of Discontinuity Infographic -
Calculus, Removable Discontinuities and Holes Infographic - Limits & Continuity Cheat Sheet
- Continuity & Types of Discontinuity Cheat Sheet
- Tangent Lines & The Derivative as a Limit Cheat Sheet
- L'Hôpital's Rule & Indeterminate Forms Cheat Sheet
- Calculus, Derivatives Worksheet
- Calculus, Series and Sequences Worksheet
- Calculus, Integrals Worksheet
- Limits & Continuity Explorer Tool
- Function Graph Explorer Tool
- L'Hopital's Rule & Indeterminate Forms Explorer Tool
- Limits & Continuity Lab Lab
- Derivatives & Tangent Lines Lab Lab
- Series & Taylor Approximation Lab Lab