Question

Limits Depending on Direction of Approach $$\lim _{x \rightarrow 0^{-}} \frac{x+1}{x}$$

Short answer

\( \lim _{x \rightarrow 0^-} \frac{x+1}{x} = -\infty \)

Step-by-step solution

Understand the Direction of the Limit

Examine the notation to determine that the limit is approached from the left side of zero, which is indicated by the negative superscript on the zero (0^-). We are interested in the behavior of the function as the value of x gets very close to but less than zero.

Simplify the Function If Possible

Review the function to see if it can be simplified before taking the limit. However, for this function, no simplifications are available that will help in finding the limit as x approaches zero from the left.

Consider the Sign of the Numerator and Denominator

As x approaches zero from the left side (negative values of x), the numerator (x+1) becomes closer to 1. However, since x is negative and very close to zero, the denominator becomes a very small negative value. Thus, the fraction becomes a positive number divided by an increasingly small negative number.

Determine the Behavior of the Function

Since we are dividing by a negative number that is getting closer and closer to zero, the fraction as a whole will become increasingly negative. This means the function will tend towards negative infinity.

Conclude the Limit

Therefore, the limit of the function as x approaches zero from the left side is negative infinity. The notation for this conclusion is \[ \lim _{x \rightarrow 0^-} \frac{x+1}{x} = -\infty \].

Key concepts

One-Sided Limits

In calculus, one-sided limits are used to describe the behavior of a function as it approaches a particular point from one direction only – either from the left or the right. This approach is essential when dealing with functions that have different behaviors on either side of a point.

For example, let's take the limit of the function \( \frac{x+1}{x} \) as \( x \) approaches zero from the left. This is denoted as \( \lim _{x \rightarrow 0^{-}} \frac{x+1}{x} \). Here, the '-' sign superscript on the zero specifies that we are considering \( x \) values that are slightly less than zero – approaching zero from the left side. The behavior of the function can be drastically different when approaching from the left as opposed to from the right, which is why the distinction is crucial. One-sided limits help in understanding discontinuities and in calculating limits at points where the function's behavior changes abruptly.

Limit Notation

Proper limit notation is important for communicating the conditions under which a limit is taken. In the expression \( \lim _{x \rightarrow c} f(x) \), the symbol \( \lim \) stands for 'limit', and the expression \( x \rightarrow c \) tells us that \( x \) approaches the value \( c \) but does not necessarily reach it. If there is an additional superscript '+' or '-', like \( x \rightarrow c^- \), it specifies the direction from which \( x \) approaches \( c \) - from the left (negative side) or from the right (positive side) respectively.

It is also necessary to understand what is meant when a limit is equated with \( +\infty \) or \( -\infty \) – this notation signifies that as \( x \) approaches \( c \) the function grows without bound in the positive or negative direction. For limitless growth or decrease, these infinity symbols become a central part of expressing the behavior of the function around specific points.

Infinite Limits

The concept of infinite limits is used when a function increases or decreases without bound as it approaches a certain value of \( x \). The limit is said to be positive infinity \( (+\infty) \) if the function grows larger and larger; it is negative infinity \( (-\infty) \) if the function decreases without limit. It does not mean that the function ever 'reaches' infinity; instead, it describes the trend in the function's behavior.

For instance, in our example \( \lim _{x \rightarrow 0^{-}} \frac{x+1}{x} = -\infty \) demonstrates that as \( x \) gets closer to zero from the left, the function's value becomes increasingly negative without limit. This unbounded decrease is a characteristic of an infinite limit in the negative direction and dictates that no matter how close \( x \) gets to zero, the function value will continue to descend.

Behavior of Functions Near Points

Analyzing the behavior of functions near points involves understanding how a function behaves as the input values \( x \) get very close to a specific point but do not necessarily reach that point. This concept is crucial for studying the continuity and discontinuity of functions, as well as for predicting the function's values near points of interest.

In the given exercise, as \( x \) approaches zero from the left, we look at the sign of the numerator and denominator to predict the behavior. Since \( x \) is negative and approaching zero, the numerator \( x+1 \) approaches \( 1 \), whereas the denominator becomes very small and negative. Thus, a positive number divided by an increasingly small negative number leads to a large negative value, indicating that the function heads towards negative infinity near the point \( x = 0 \) from the left side.

This information about the function's behavior near specific points equips us with the much-needed understanding to anticipate and explain why the function acts a certain way around those points, which is crucial for successfully grasining calculus concepts.