Problem 4 - Calculus Single Variable

Question

In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow-1+}(x+1)^{-1} $$

Step-by-Step Solution

Answer

The limit does not exist as a finite number; it diverges to \(+\infty\).

Step 1: Understanding the Limit

The given problem asks for the limit of the function \((x+1)^{-1}\) as \(x\) approaches \(-1\) from the right (positive side). We recognize this as a one-sided limit, specifically a right-hand limit.

Step 2: Function Behavior near the Point

Consider the function \((x+1)^{-1}\). As \(x\) gets closer and closer to \(-1\) from the right, the expression \(x+1\) approaches zero from the positive side. Hence, the denominator approaches zero positively.

Step 3: Analyzing the Limit Expression

When the denominator approaches zero from the positive side in the expression \((x+1)^{-1}\), the fraction becomes very large. Therefore, as \(x\to -1^+\), \((x+1)^{-1}\) tends towards positive infinity.

Step 4: Conclusion on Limit Existence

Since the value of \((x+1)^{-1}\) increases without bound as \(x\) approaches \(-1\) from the right, the limit does not exist in the finite sense. It diverges to \(+\infty\).

Key Concepts

One-Sided Limits

When we talk about one-sided limits in calculus, we're interested in how a function behaves as it approaches a particular x-value from only one direction—left or right. The concept is critical in understanding the function's behavior right at the edge of a point.
A right-handed limit, noted as \( \lim_{x \to c^+} f(x) \), observes the behavior as \( x \) approaches \( c \) from values larger than \( c \). Similarly, a left-handed limit, \( \lim_{x \to c^-} f(x) \), checks from values smaller than \( c \).

• Whether these limits exist or not can tell us a lot about the nature of the function at that point.
• If both limits exist and are the same, the two-sided limit \( \lim_{x \to c} f(x) \) is said to exist.

In our exercise, we're specifically looking at a right-sided limit— \( x \to -1^+ \)—meaning \( x \) is approaching -1 from the positive side. Understanding this helps us grasp how functions can behave differently when approached from certain directions.

Behavior of Rational Functions

Rational functions are quotients of two polynomials. When encountering rational functions, a lot hinges on the denominator. If it shrinks towards zero, the value of the function can grow very large, either negatively or positively, depending on the context. This behavior is key to understanding limits involving rational expressions.

• In \((x+1)^{-1}\), the denominator \(x+1\) dictates the behavior as we change x.
• When we approach -1 from the right, \(x+1\) tips towards zero but remains positive.
• The fraction \((x+1)^{-1}\) responds by increasing towards very large positive values.

This property of rational functions, especially when the denominator crosses zero, is often critical in determining the nature of limits like this one. Knowing this can help predict when and where a function might skyrocket towards infinity or plummet downwards.

Infinite Limits

Infinite limits occur when the value of a function grows without bounds as it heads towards a certain point. These types of limits don't result in real numbers; instead, they approach infinity or negative infinity. Understanding when a function tends towards infinity is vital in calculus as it informs us about points of unbounded behavior.

• In our exercise, as \(x\) marches towards -1 from the right, \((x+1)^{-1}\) surges towards infinity.
• This reflects an infinite limit, showing the function climbs endlessly as the denominator hovers around zero.

Infinite limits help us describe and understand singular points where functions become unrestrained. While these limits don't exist in the conventional sense, they offer insight into the more extreme behavior of mathematical expressions.