A Function $\mathrm { f } ( \mathrm { x } )$

question 115

Multiple Choice

A function $\mathrm { f } ( \mathrm { x } )$ , a point $\mathrm { c }$ , the limit of $\mathrm { f } ( \mathrm { x } )$ as $x$ approaches $\mathrm { c }$ , and a positive number $\varepsilon$ is given. Find a number $\delta > 0$ such that for all $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$ .
- $$f ( x ) = m x + b , m > 0 , L = \frac { m } { 5 } + b , c = \frac { 1 } { 5 } , \text { and } \varepsilon = c > 0$$

• A) \(\delta = \frac { c } { 5 }\)
• B) \(\delta = \frac { c } { m }\)
• C) \(\delta = \frac { 5 } { m }\)
• D) \(\delta = \frac { 1 } { 5 } + \frac { c } { m }\)

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Definitions:

Independent Samples

Samples in which the sample values of one group do not depend on the values of another, often used in hypothesis testing to compare two groups.

Normal Populations

Populations that follow a normal distribution, characterized by its bell-shaped curve, where the mean, median, and mode are all equal.

Equal Variances

Assuming that two or more populations have the same variance, often an assumption in statistical tests comparing populations.

Sample Variances

Sample variances refer to the measurement of the variance within a sample from a population, providing an estimate of the population variance.