Question

Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 1^{+}} \frac{5}{1-x} $$

Short answer

The limit of the given function as \(x\) approaches \(1^{+}\) is \(-\infty\).

Step-by-step solution

Identify the limit

The given expression is \(\lim _{x \rightarrow 1^{+}} \frac{5}{1-x}\). Here, the function \(f(x) = \frac{5}{1-x}\), and \(x\) approaches \(1^{+}\), i.e., from the right side of 1.

Analyze the function behavior

Firstly, replace \(x\) in the function \(f(x)\) with values slightly greater than 1, say, 1.00001 (closer to 1 from the right side) and see how the function behaves. The function gives -500000. The result seems to be heading towards negative infinity. This because as \(x\) gets closer to 1, the denominator becomes very close to zero, making the overall expression very large in magnitude. Moreover, since \(x\) is greater than 1, \(1-x\) is negative, which leads to the whole expression to be negative.

Mathematical calculation of the limit

The function has the form \(\frac{a}{b}\) where \(b\) approaches 0 from the right and a is positive. For such a function, if \(b \rightarrow 0^{+}\), then \(\frac{a}{b} \rightarrow -\infty\), and if \(b \rightarrow 0^{-}\), then \(\frac{a}{b} \rightarrow \infty\). In this case, 1-x as \(x \rightarrow 1^{+}\) implies that, 1−x \rightarrow 0^{+}\. Hence, by the above rules, \(\lim _{x \rightarrow 1^{+}} \frac{5}{1-x} = -\infty\)

Key concepts

One-Sided Limits

One-sided limits help us understand the behavior of a function as it approaches a particular value from only one direction, either the right or the left. In this exercise, we are dealing with the one-sided limit \(\lim _{x \rightarrow 1^{+}} \frac{5}{1-x}\), which means we are looking at how the function behaves as \(x\) gets closer to 1 from values greater than 1. This is signified by the superscript plus sign \(1^{+}\).

Understanding one-sided limits is key because functions can behave differently on either side of a number. For example, as in our problem, approaching from the right (\(1^{+}\)) might yield results that differ dramatically compared to approaching from the left (\(1^{-}\)). This can help solve more complex functions where the limits may not be obvious by just looking at \(x \rightarrow c\) without specifying a direction.

Infinity

In mathematical terms, infinity represents a number larger than any real number. It is a concept rather than a fixed value, which means that while we cannot reach infinity, we can get arbitrarily close to it. In the given problem, analyzing \(\lim _{x \rightarrow 1^{+}} \frac{5}{1-x}\), we encounter a scenario where the function's output becomes infinitely large (negatively, in this case).

As \(x\) approaches 1 from the right, the denominator \(1-x\) becomes very small but negative, driving the fraction's value towards negative infinity. This is why understanding the concept of infinity is crucial – it allows us to express the unbounded behavior of functions rather than trying to compute an unreachable numeric result.

Function Behavior

When we talk about function behavior near a certain point, we're examining what happens to the value of the function as the input gets infinitesimally close to a particular point. In our example, the function \(f(x) = \frac{5}{1-x}\) is examined as \(x\) approaches 1 from numbers slightly greater than 1. This helps us anticipate the behavior of the function without actually reaching that exact point.

By substituting very close values like \(x = 1.00001\), we saw that the output became extremely negative. This illustrates that the function behaves by tending toward negative infinity on the right side of 1. Recognizing these patterns and behaviors helps predict and understand complex functions, enabling us to draw conclusions about limit values.

Analytic Methods

Analytic methods refer to the process of using algebraic techniques and mathematical reasoning to evaluate limits. These methods often involve simplifying expressions and recognizing patterns of behavior as a variable approaches a particular value. In this problem, we employ the fundamental understanding of how fractions behave when denominators approach zero.

Knowing that \(\frac{a}{b}\) becomes extremely large negative when \(b\) approaches a very small positive value (\(0^{+}\)), we can confidently conclude that \(\lim _{x \rightarrow 1^{+}} \frac{5}{1-x} = -\infty\). This contrasts with numerical approaches which may involve plugging in values but might not sufficiently capture the asymptotic behavior. The power of analytic methods lies in their ability to generalize and offer insights that are not immediately obvious from numeric or graph-based analyses.