Mr. IntegralTHE WHOLE UNIT · ONE REFERENCE
Limits & Continuity
Cheat sheet.
The key rules, formulas and reminders from all 5 topics, gathered into reference cards.
Key formulas, conditions and traps · Read down each column.
Limits and One-Sided Behavior
Core rule
A finite two-sided limit exists exactly when its two one-sided limits agree. The value at the point need not equal the limit.
Watch for
is an indeterminate form. Simplify on a punctured neighborhood; do not substitute a value into a cancelled original denominator. Tables and plots suggest limits but do not prove them.
Continuity and Piecewise Functions
Core rule
Continuity at requires a defined value, an existing finite limit, and equality between them. Check one-sided limits at piecewise boundaries.
Watch for
A point-value change repairs only a removable discontinuity. Continuity does not imply differentiability; differentiability does imply continuity.
Intermediate Values and Root Bracketing
Core rule
For continuous on , every value between the endpoint outputs is attained. Opposite signs guarantee an interior root.
Watch for
No uniqueness follows without another argument. A discontinuity invalidates the guarantee. Bisection midpoint error after halvings is at most .
Infinite Limits and Asymptotes
Core rule
Vertical asymptotes require an infinite one-sided limit. Horizontal asymptotes are finite limits at an infinite end. Slant asymptotes satisfy .
Watch for
Cancel common factors before identifying poles. Inspect one-sided signs. A graph may cross an asymptote.
Factoring, Rationalizing and Squeezing Limits
Core rule
Use equality on nearby non-target points for factoring or rationalization. If nearby and , then .
Watch for
Angles must be in radians. Bounds must hold throughout a neighborhood. is a prompt for analysis, not an answer.
Limits and One-Sided Behavior
2 reference blocks
Core rule
A finite two-sided limit exists exactly when its two one-sided limits agree. The value at the point need not equal the limit.
Watch for
is an indeterminate form. Simplify on a punctured neighborhood; do not substitute a value into a cancelled original denominator. Tables and plots suggest limits but do not prove them.
Continuity and Piecewise Functions
2 reference blocks
Core rule
Continuity at requires a defined value, an existing finite limit, and equality between them. Check one-sided limits at piecewise boundaries.
Watch for
A point-value change repairs only a removable discontinuity. Continuity does not imply differentiability; differentiability does imply continuity.
Intermediate Values and Root Bracketing
2 reference blocks
Core rule
For continuous on , every value between the endpoint outputs is attained. Opposite signs guarantee an interior root.
Watch for
No uniqueness follows without another argument. A discontinuity invalidates the guarantee. Bisection midpoint error after halvings is at most .
Infinite Limits and Asymptotes
2 reference blocks
Core rule
Vertical asymptotes require an infinite one-sided limit. Horizontal asymptotes are finite limits at an infinite end. Slant asymptotes satisfy .
Watch for
Cancel common factors before identifying poles. Inspect one-sided signs. A graph may cross an asymptote.
Factoring, Rationalizing and Squeezing Limits
2 reference blocks
Core rule
Use equality on nearby non-target points for factoring or rationalization. If nearby and , then .
Watch for
Angles must be in radians. Bounds must hold throughout a neighborhood. is a prompt for analysis, not an answer.