Concept 03 / 12Limits, Continuity and Differentiability
L'Hopital's Rule, Series Expansions and the Sandwich Theorem
When and how to apply L'Hopital's rule to 0/0 and infinity/infinity forms, Maclaurin expansions of standard functions for evaluating limits and finding unknown constants, and the sandwich theorem for oscillating and greatest-integer expressions.
L'Hopital's rule. If and as (or both tend to ), and are differentiable near with near , then
,
provided the limit on the right exists (or is ). Differentiate the numerator and denominator separately; this is not the quotient rule. The rule can be applied repeatedly as long as the form remains or . Check the form before each application: applying it to a determinate form gives wrong answers. For example, , but 'differentiating' gives .
Other indeterminate forms must be converted first. : write . : combine into a single fraction. Exponential forms: take logarithms.
When the right-hand limit fails to exist, L'Hopital says nothing. For as , the derivative ratio oscillates, but the original limit is clearly . Use the rule as a tool, not a definition.
Series expansions (Maclaurin). Near :
for
and
Substituting expansions is often faster than repeated L'Hopital, especially for expressions like , and it is the method of choice when the question asks for constants that make a limit finite. The rule: expand every function up to the order of the lowest non-cancelling power in the denominator.
Finding unknown constants. If a limit of the form is finite, the coefficients of in the expansion of must all be zero; the limit is then the coefficient of . Each vanishing condition gives an equation for the unknowns.
The sandwich (squeeze) theorem. If near and , then . It handles expressions that oscillate or involve , where algebra and L'Hopital fail. The figure shows trapped between and : since both bounds tend to , so does the function, even though itself has no limit. The general principle: a bounded function times a function that tends to tends to . For the greatest integer function use ; for example, and .
Key Takeaways:
- L'Hopital: or only; differentiate top and bottom separately.
- Convert and to a quotient first.
- Expansions: , , , , .
- For a finite : coefficients of lower powers in vanish.
- Squeeze: bounded (something ) .
- gives squeeze bounds for greatest-integer limits.
- Pitfall: applying L'Hopital when the form is not indeterminate.
Brain-Teaser 1: Evaluate .
Solution: By series: . So the limit is . Alternatively, , so the expression is . A common error is to replace and both by and conclude the limit is ; substituting first-order approximations inside a difference loses exactly the terms that matter.
Brain-Teaser 2: Find and such that .
Solution: Expand: . For a finite limit, the coefficient must vanish: . The limit is then the coefficient: . Solving: , so , giving , and . Series expansion turns 'find the constants' into two linear equations.
Brain-Teaser 3: Evaluate and .
Solution: For any real , . With and multiplying by : . Both bounds tend to , so the limit is . For the second, as the same argument gives . As , multiplying by reverses the inequalities: , which also squeezes to . So both limits are . Multiplying an inequality by a negative quantity reverses it, and forgetting this is the usual error on the left side.
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