Question

Definition of one-sided infinite limits We write \(\lim _{x \rightarrow a^{+}} f(x)=-\infty\) if for any negative number \(N\), there exists \(\delta>0\) such that $$ f(x)

Short answer

Question: Provide the formal definitions of one-sided infinite limits for the following cases: a. \(\lim_{x \rightarrow a^{+}} f(x) = \infty\) b. \(\lim_{x \rightarrow a^{-}} f(x) = -\infty\) c. \(\lim_{x \rightarrow a^{-}} f(x) = \infty\) Answer: a. \(\lim_{x \rightarrow a^{+}} f(x) = \infty\) if for any positive number \(M\), there exists \(\delta > 0\) such that \(f(x) > M\) whenever \(0 < x - a < \delta\). b. \(\lim_{x \rightarrow a^{-}} f(x) = -\infty\) if for any negative number \(N\), there exists \(\delta > 0\) such that \(f(x) < N\) whenever \(0 < a - x < \delta\). c. \(\lim_{x \rightarrow a^{-}} f(x) = \infty\) if for any positive number \(M\), there exists \(\delta > 0\) such that \(f(x) > M\) whenever \(0 < a - x < \delta\).

Step-by-step solution

(a. Formal definition for \(\lim_{x \rightarrow a^{+}} f(x) = \infty\))

We can write the formal definition for \(\lim_{x \rightarrow a^{+}} f(x) = \infty\) as follows: \(\lim_{x \rightarrow a^{+}} f(x) = \infty\) if for any positive number \(M\), there exists \(\delta > 0\) such that $$ f(x) > M \text{ whenever } 0 < x - a < \delta $$

(b. Formal definition for \(\lim_{x \rightarrow a^{-}} f(x) = -\infty\))

We can write the formal definition for \(\lim_{x \rightarrow a^{-}} f(x) = -\infty\) as follows: \(\lim_{x \rightarrow a^{-}} f(x) = -\infty\) if for any negative number \(N\), there exists \(\delta > 0\) such that $$ f(x) < N \text{ whenever } 0 < a - x < \delta $$

(c. Formal definition for \(\lim_{x \rightarrow a^{-}} f(x) = \infty\))

We can write the formal definition for \(\lim_{x \rightarrow a^{-}} f(x) = \infty\) as follows: \(\lim_{x \rightarrow a^{-}} f(x) = \infty\) if for any positive number \(M\), there exists \(\delta > 0\) such that $$ f(x) > M \text{ whenever } 0 < a - x < \delta $$

Key concepts

Limit Definitions

Understanding the concepts behind limits is fundamental in calculus. A limit describes the value that a function approaches as the input (or independent variable) approaches a certain value. When we talk about the limit of a function as 'x' approaches a particular value 'a', we are interested in what value 'f(x)' gets closer to, not necessarily reaching it. This concept helps us deal with behavior near points that might not be clearly defined or are unreachable by the function itself.

There are various types of limits, but they all serve the same purpose: to give us a way to talk about the value of a function's output based on its input. A limit exists if 'f(x)' approaches a specific number as 'x' approaches 'a'. Infinite limits, in particular, describe the behavior of a function as it increases or decreases without bound near a particular point.

One-Sided Limits

One-sided limits take the concept of limits further by focusing on the behavior of a function as the independent variable approaches a specific value from one direction: either from the left (denoted as \( x \rightarrow a^- \) ) or from the right (denoted as \( x \rightarrow a^+ \) ). These limits are especially useful when a function behaves differently depending on the direction from which 'x' approaches 'a'.

For example, the one-sided limit as 'x' approaches 'a' from the right ( \( \lim_{x \rightarrow a^{+}} f(x) \) ) is the value that 'f(x)' is approaching as x gets infinitely close to 'a' from values larger than 'a'. Conversely, the limit as 'x' approaches 'a' from the left ( \( \lim_{x \rightarrow a^{-}} f(x) \) ) is the value that 'f(x)' is approaching as x gets infinitely close to 'a' from values smaller than 'a'. These concepts help us understand and describe the behavior of functions at specific points, even if the overall function has different behavior on either side of that point.

Calculus

Calculus is the branch of mathematics that deals with continuous change. It is divided mainly into two areas: differential calculus and integral calculus. Differential calculus concerns the concept of instantaneous rate of change, which is fundamentally about limits — specifically, the limit of the average rate of change of a function as the interval over which we're measuring becomes infinitesimally small. Calculus helps us explore and solve problems in physics, engineering, economics, and many other fields by providing tools to model and analyze dynamic systems.

Integral calculus, on the other hand, deals with accumulations of quantities, such as areas under curves and the total accumulation of a quantity over a range of values. Both concepts, although diverging in application, intertwine at a deep mathematical level through the Fundamental Theorem of Calculus, which links the concept of derivatives to integrals.

Limits Approaching Infinity

When dealing with limits approaching infinity, we're often looking at the behavior of a function as 'x' grows very large or decreases without bound. The notation \( \lim_{x \rightarrow \infty} f(x) = L \) indicates that as 'x' becomes indefinitely large, the function 'f(x)' approaches the value 'L'. The situation is similar when 'x' approaches negative infinity. In the context of the exercise, when a function approaches positive infinity as 'x' approaches a from the right ( \( \lim_{x \rightarrow a^{+}} f(x) = \infty \) ), the function's values become unbounded in the positive direction near 'a'.

Similarly, a function may approach positive infinity as 'x' approaches a from the left ( \( \lim_{x \rightarrow a^{-}} f(x) = \infty \) ). This signifies that as 'x' comes from values smaller than 'a', 'f(x)' grows without restriction. Such behaviors highlight how functions can exhibit dramatic and unbounded change near certain points, and these concepts are essential in understanding the infinity in a mathematical context.