Final answer:
Limits in mathematics involve approaching a point, not necessarily reaching it, and the concept of infinity is not a number but rather a mathematical idea. Continuous probability functions are characterized by the property that the area under the function represents probability, with the total area being 1.
Step-by-step explanation:
Understanding limits and continuous probability functions is crucial in mathematics, particularly in calculus and probability theory. Let's review each statement about limits:
- (a) False. If limx→a f(x)=L, it does not mean f(x) lies in the interval (a-δ, a+δ), but rather that for each ε > 0, there exists a δ > 0 such that if 0 <|x-a|<δ, then |f(x)-L|<ε.
- (b) False. f(a) = L is possible, but it does not affect the limit as the limit concerns the values approaching a, not at a itself.
- (c) True. If limx→a f(x)=[infinity], then f(x) is greater than any large number M for x values sufficiently close to a.
- (d) False. Infinity is a concept, not a number.
- (e) False. If x is within δ of a, this does not necessarily mean the limit of f(x) is finite—it can also be infinite or undefined.
When speaking of continuous probability functions, we equate the area under the probability density function with the probability. Important properties of such distributions include:
- Probability of a specific point in a continuous distribution is always zero because the area at a single point is zero.
- The total area under a continuous probability density function across its entire range is equal to 1.