Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{x \rightarrow 0^{-}} \frac{1+\cos x}{\sin x}\)

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Identify the Limit Direction

We begin by recognizing that the limit is approached as \( x \) approaches zero from the left (indicated by \( x \to 0^- \)). This means that \( x \) takes negative values close to zero.

Step 2: Substitute the Trigonometric Functions Near Zero

We substitute values very close to zero in the trigonometric functions:- Near zero, \( \cos(x) \approx 1 \).- Near zero, \( \sin(x) \approx x \).

Step 3: Rewrite the Limit Expression Using Approximations

Substitute these approximations into the limit expression:\[\lim_{x \to 0^-} \frac{1 + \cos x}{\sin x} \approx \lim_{x \to 0^-} \frac{1 + 1}{x} = \lim_{x \to 0^-} \frac{2}{x}\]

Step 4: Evaluate the Behavior of the Limit Expression

Now evaluate the rewritten limit expression:\(\lim_{x \to 0^-} \frac{2}{x}\). Since \(x\) is approaching zero from the left (negative values), \(\frac{2}{x}\) will approach \(-\infty\).

Step 5: Conclusion of the Limit

Conclude based on the previous step that:\[\lim _{x \rightarrow 0^-} \frac{1+ ext{cos} x}{ ext{sin} x} = -\infty\]

Key Concepts

Trigonometric Limits

Trigonometric limits involve finding the behavior of expressions that include trigonometric functions as they approach specific points. These points often involve angles that make the trigonometric functions undergo special transformations. For instance, at zero, the value of many trigonometric functions simplifies, making limit calculations more manageable. In this particular example, we consider trigonometric functions like \( \cos x \) and \( \sin x \), noting that as \( x \) approaches zero:

• \( \cos(x) \approx 1 \)
• \( \sin(x) \approx x \)

These approximations simplify the original limit expression, helping us conclude the behavior of the function as it nears zero.

Limit Evaluation

Limit evaluation is the process of determining the value that a function or expression approaches as the input approaches some point. In calculus, this often involves manipulating expressions to reveal their behavior. Here, we start by substituting near-zero approximations for sine and cosine, transforming the initial limit into a simpler form. This aids in approximating the limit value. By using the fact that near zero, \( \cos x \approx 1 \) and \( \sin x \approx x \), we rewrite the limit expression:

• \[ \lim_{x \to 0^-} \frac{1 + \cos x}{\sin x} \approx \lim_{x \to 0^-} \frac{1 + 1}{x} = \lim_{x \to 0^-} \frac{2}{x} \]

Through this technique, evaluating complicated limits becomes more straightforward, building foundational skills for tackling more intricate calculus problems.

Asymptotic Behavior

Asymptotic behavior describes how functions behave as inputs get very large or very small. In limits, it often concerns how functions grow or shrink indefinitely, such as approaching infinity. In the given exercise, as \( x \) approaches zero from the left, \( \frac{2}{x} \) tends toward \(-\infty\), meaning its size increases negatively without bound:

• The function value \( \frac{2}{x} \) grows negatively large as \( x \to 0^- \).
• The minus indicates the direction of growth, towards negative infinity.

Understanding asymptotic behavior is essential in calculus as it highlights limits where the traditional numeric answer is not defined, yet the behavior is still described.

Single-Sided Limits

Single-sided limits involve approaching a specific value from only one side, either from the left (negative direction) or the right (positive direction). It gives insight into the function's behavior as it nears a point from a particular side, which can differ from the behavior when approaching from the other side. Here, the limit is approached from the left, denoted by \( x \to 0^- \). This kind of limit is useful when studying scenarios where the function has discontinuities or different definitions depending on sign:

• Approaching zero from the left ensures \( x \) stays in the negative area close to zero.
• The outcomes can confirm whether the function has an abrupt change or endless behavior when nearing specific values from one direction.

Single-sided limits provide a detailed analysis of the function's behavior at critical points, especially when symmetrical analysis isn't possible due to differences in behavior from either side.