Use the precise definition of infinite limits to prove the following limits.


$\underset{x\to 0}{\lim }(\frac{1}{x^4}-\sin \left(x\right))=\infty$

Verified step by step guidance

1. Step 1: Understand the definition of an infinite limit. The limit \( \lim_{x \to a} f(x) = \infty \) means that for every positive number \( M \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( f(x) > M \).
2. Step 2: Consider the function \( f(x) = \frac{1}{x^4} - \sin(x) \). As \( x \to 0 \), \( \frac{1}{x^4} \to \infty \) and \( \sin(x) \to 0 \). Therefore, \( f(x) \to \infty \).
3. Step 3: For a given \( M > 0 \), we need to find \( \delta > 0 \) such that if \( 0 < |x| < \delta \), then \( \frac{1}{x^4} - \sin(x) > M \).
4. Step 4: Since \( \sin(x) \) is bounded, \( |\sin(x)| \leq |x| \). Therefore, \( \frac{1}{x^4} - \sin(x) \geq \frac{1}{x^4} - |x| \).
5. Step 5: Choose \( \delta \) small enough such that \( \frac{1}{x^4} - |x| > M \) for \( 0 < |x| < \delta \). This ensures that \( \frac{1}{x^4} - \sin(x) > M \), proving the limit is infinite.

Key Concepts

Infinite Limits

Infinite limits describe the behavior of a function as the input approaches a certain value, where the function's output grows without bound. Specifically, if the limit of a function as x approaches a value results in infinity, it indicates that the function increases indefinitely in that vicinity. Understanding infinite limits is crucial for analyzing functions that exhibit vertical asymptotes or unbounded growth.

Sine Function Behavior

The sine function, denoted as sin(x), oscillates between -1 and 1 for all real numbers x. As x approaches 0, sin(x) can be approximated by its Taylor series expansion, which shows that sin(x) behaves like x near zero. This property is essential for evaluating limits involving sin(x) and understanding how it interacts with other functions, particularly in the context of limits approaching zero.

Limit Definition and Evaluation

The precise definition of a limit involves the concept of approaching a value as closely as desired. To prove limits, one often uses algebraic manipulation, substitution, or the epsilon-delta definition. In this case, evaluating the limit requires analyzing the expression as x approaches 0, particularly focusing on the dominant term in the expression, which is crucial for determining the limit's behavior.