Math Problem Statement
lim/x→0 sin x - tan x / x^3
Solution
To solve the limit
we'll analyze the expression in the numerator, , and rewrite it to make it easier to evaluate.
Step 1: Rewrite in terms of and
We know that , so we can rewrite the expression as:
Step 2: Simplify
Using the identity , we can rewrite the expression as:
Step 3: Expand and using Taylor series
For small , we can use the Taylor series expansions:
Substitute these approximations into the expression:
Step 4: Simplify the expression
Multiplying out terms, we get:
Step 5: Take the limit as
As , the term goes to zero, leaving us with:
Final Answer
Would you like additional details on Taylor series expansion, or help with any questions about this process?
- What is the Taylor series and how does it help in evaluating limits?
- How can you approximate and for small ?
- Why do higher-order terms vanish when taking limits?
- What other methods can be used for limits involving trigonometric functions?
- How does Taylor expansion compare with L'Hôpital's Rule for solving limits?
Tip: For limits involving indeterminate forms like , Taylor expansions can simplify the problem by approximating functions around the point of interest.
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometric Functions
Taylor Series Expansion
Formulas
lim_{x→0} (sin x - tan x) / x^3
sin x ≈ x - x^3/6 + O(x^5)
cos x - 1 ≈ -x^2/2 + O(x^4)
Theorems
Taylor Series Expansion
Limit Laws
Suitable Grade Level
Grades 11-12
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