Limits of functions and continuity¶
Part 3 of the lecture notes. Limits, asymptotes, standard limits, continuity, zeros theorem, Weierstrass.
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1. Limits of functions, asymptotes and continuity
Sequential definition of limit
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2. Computing limits of functions
Computing limits of functions · Change of variable theorem for limits
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3. Limits of polynomials and rational functions
Limits of polynomials at … · Limits of rational functions at … · Limits of quotients of sums of powers with rational exponents
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4. Continuous functions
Algebra of continuous functions theorem · Continuity of elementary functions theorem · Continuity of composite functions theorem
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5. Comparison of infinities
Comparison of infinities for functions
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6. Fundamental limits and asymptotic estimates
Fundamental limits · Asymptotic estimates · Asymptotic estimates and graphs
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7. Asymptotic expansions
The “little …” symbol and asymptotic expansions · Asymptotic expansions derived from the fundamental limits
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8. Intermediate zero theorem and bisection method
Zeros of a function · Bisection method
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9. Weierstrass theorem and intermediate value theorem
Weierstrass theorem · Intermediate value theorem · Existence theorem for the …-th root
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10. Monotone functions on an interval and invertibility
Monotone functions on an interval · Continuity and invertibility