Calculus I · Limits and Continuity · lesson
Scaled Sine and Tangent Limits
Learning objectives
Rewrite trigonometric limits so that a factor has the standard form or .
Scaled Sine and Tangent Limits
The substitution pattern
If , then whenever . We want to create the denominator that matches the sine argument.
Evaluate
Show worked solution
The sine argument is , but the denominator is only . Multiply and divide by :
As , , so
Therefore,
Keep the outside constant
Evaluate
Show worked solution
Create in the denominator:
The standard factor approaches , so
After the explanation
Use the section idea
Can the expression be rewritten around a known small-angle limit, with every scaling factor accounted for?
The fundamental sine limit is a reusable local shape: other trigonometric limits work when you expose that shape through identities and scaling.
Look for a bounded oscillation times a shrinking factor, or rewrite the expression into sine-over-angle factors whose arguments match their denominators.
The sine function is not equal to its angle; their ratio merely approaches one near zero, and that statement requires radian measure.
Mastery means you can mark every scaling factor before simplifying and can explain where the Squeeze Theorem enters the argument.
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