Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
- 8. Definite Integrals3h 25m
4. Applications of Derivatives
Differentials
Problem 4.7.54
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→1⁻ (1-x) tan πx/2
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1
Identify the limit to evaluate: lim_{x→1⁻} (1-x) tan(πx/2). As x approaches 1 from the left, (1-x) approaches 0 and tan(πx/2) approaches infinity, creating a 0 * ∞ form.
Rewrite the expression to apply l'Hôpital's Rule: lim_{x→1⁻} (1-x) tan(πx/2) can be rewritten as lim_{x→1⁻} (tan(πx/2) / (1/(1-x))). This transforms the limit into a 0/0 form.
Differentiate the numerator and denominator separately: The derivative of the numerator, tan(πx/2), is (π/2)sec²(πx/2) and the derivative of the denominator, 1/(1-x), is 1/(1-x)².
Apply l'Hôpital's Rule: Substitute the derivatives back into the limit: lim_{x→1⁻} (π/2)sec²(πx/2) / (1/(1-x)²).
Evaluate the new limit as x approaches 1 from the left, simplifying the expression to find the limit value.
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