- 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
1. Limits and Continuity
Finding Limits Algebraically
Problem 4f
Textbook Question
Limits and Continuity
Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.
f. [ƒ(x) • cos x ] / x―1

1
First, identify the limit expression you need to evaluate: \( \lim_{x \to 0} \frac{f(x) \cdot \cos x}{x - 1} \).
Recognize that you can use the limit properties to separate the limit of a product: \( \lim_{x \to 0} f(x) \cdot \lim_{x \to 0} \cos x \).
Since \( \lim_{x \to 0} f(x) = \frac{1}{2} \) and \( \lim_{x \to 0} \cos x = \cos(0) = 1 \), substitute these values into the expression.
Now, evaluate the denominator \( x - 1 \) as \( x \to 0 \), which becomes \( 0 - 1 = -1 \).
Combine the results: \( \frac{\frac{1}{2} \cdot 1}{-1} \). Simplify this expression to find the limit.
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