Question

Give explicit definitions for the following "unsigned infinity" limit statements: (a) \(\lim _{x \rightarrow p} f(x)=\infty\) (b) \(\lim _{x \rightarrow p^{+}} f(x)=\infty\); (c) \(\lim _{x \rightarrow \infty} f(x)=\infty\)

Short answer

Each statement describes a behavior of \( f(x) \) becoming arbitrarily large as \( x \) approaches \( p \), \( p^+ \), or \( \infty \).

Step-by-step solution

Understanding Limit Statement (a)

For the statement \( \lim_{x \rightarrow p} f(x) = \infty \), it means as \( x \) approaches \( p \) from either direction, the values of \( f(x) \) become arbitrarily large. Explicitly, for any given large number \( M > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - p| < \delta \), then \( f(x) > M \).

Understanding Limit Statement (b)

For the statement \( \lim_{x \rightarrow p^{+}} f(x) = \infty \), it means as \( x \) approaches \( p \) specifically from the right (or from values greater than \( p \)), the values of \( f(x) \) become arbitrarily large. Specifically, for any large number \( M > 0 \), there is a \( \delta > 0 \) such that when \( 0 < x - p < \delta \), then \( f(x) > M \).

Understanding Limit Statement (c)

For the statement \( \lim_{x \rightarrow \infty} f(x) = \infty \), it means as \( x \) increases without bound, the values of \( f(x) \) also increase without bound. More formally, for any large number \( M > 0 \), there exists a number \( N > 0 \) such that for all \( x > N \), \( f(x) > M \).

Key concepts

limit definitions

The concept of limits is fundamental in calculus and mathematical analysis. Limits help us understand how a function behaves as its input approaches a particular value or as it goes to infinity. When defining limits, we often describe what happens as we get extremely close to a point or as a variable grows indefinitely.- **Limit Statement (a):** The expression \( \lim _{x \rightarrow p} f(x)=\infty \) means that as \( x \) gets closer to a specific value \( p \), from either side, the value of \( f(x) \) becomes extremely large. Imagine being able to pick a number, no matter how huge, and \( f(x) \) will eventually surpass it as \( x \) nears \( p \). - **Limit Statement (b):** The expression \( \lim_{x \rightarrow p^{+}} f(x)=\infty \) specifies that \( x \) is approaching \( p \) from the right, meaning from values greater than \( p \). \( f(x) \) will grow without bound, becoming larger and larger.- **Limit Statement (c):** The expression \( \lim _{x \rightarrow \infty} f(x)=\infty \) focuses on what happens as \( x \) itself grows larger. In this case, as \( x \) increases, \( f(x) \) also continues to grow indefinitely.

mathematical analysis

Mathematical analysis is a branch of mathematics that deals with the limits, continuity, and smoothness of functions. It provides tools for examining and understanding the behavior of functions at a very detailed level. - **Understanding Continuity:** Within mathematical analysis, continuity ensures that small changes in an input lead to small changes in the output. For limits involving infinity, the focus is on how the function behaves as it stretches to endless value. - **Precision in Analysis:** Mathematical analysis requires precise definitions and rigorous proof techniques. When discussing limits, it involves dealing with arbitrarily small or large numbers and ensuring that the definition holds under these circumstances. - **Impact on Function Behavior:** By understanding limits, one can make predictions about the overall behavior of a function, determine if it's close to smooth, and how it reacts as it reaches extreme values.

unsigned infinity

Unsigned infinity in mathematics refers to concepts that describe growth or extent without direction, just as simple infinity. When we say a function's limit approaches infinity, it involves this concept of unsigned, or limitless growth.- **No Upper Bound:** In limit expressions like \( \lim_{x \rightarrow p} f(x) = \infty \), unsigned infinity means there's no upper bound on the output as input nears some point. The function just keeps increasing.- **Behavioral Understanding:** By understanding unsigned infinity, we can grasp how certain processes or functions move toward boundlessness. This helps when analyzing the growth knowledgeably.- **Application in Graphs:** On a function graph, unsigned infinity is represented by a curve that keeps climbing as \( x \) progresses towards specific values or over infinity. This visual representation helps us understand the rate and nature of this boundless growth.