Question

Find (a) \(\lim _{x \rightarrow \infty} x^{2 k}\) (b) \(\lim _{x \rightarrow-\infty} x^{2 k}\) (c) \(\lim _{x \rightarrow \infty} x^{2 k+1}\) (d) \(\lim _{x \rightarrow-\infty} x^{2 k+1}\) (k=positive integer) (e) \(\lim _{x \rightarrow \infty} \sqrt{x} \sin x\) (f) \(\lim _{x \rightarrow \infty} e^{x}\)

Short answer

Question: Find the limits of the following functions as x approaches infinity or negative infinity: (a) \(x^{2k}\) as x approaches infinity (b) \(x^{2k}\) as x approaches negative infinity (c) \(x^{2k+1}\) as x approaches infinity (d) \(x^{2k+1}\) as x approaches negative infinity (e) \(\sqrt{x} \sin x\) as x approaches infinity (f) \(e^{x}\) as x approaches infinity Answer: (a) The limit of \(x^{2k}\) as x approaches infinity is infinity. (b) The limit of \(x^{2k}\) as x approaches negative infinity is infinity. (c) The limit of \(x^{2k+1}\) as x approaches infinity is infinity. (d) The limit of \(x^{2k+1}\) as x approaches negative infinity is negative infinity. (e) The limit of \(\sqrt{x} \sin x\) as x approaches infinity does not exist. (f) The limit of \(e^{x}\) as x approaches infinity is infinity.

Step-by-step solution

(a) Limit as x approaches infinity for \(x^{2k}\)

To find the limit as x approaches infinity for \(x^{2k}\), we simply need to analyze the behavior of the function when x becomes very large. Since \(x^{2k}\) is a polynomial function with a positive even power, it will always be positive, and its value will increase as x becomes larger. Therefore, the limit as x approaches infinity for \(x^{2k}\) is infinity.

(b) Limit as x approaches negative infinity for \(x^{2k}\)

To find the limit as x approaches negative infinity for \(x^{2k}\), we need to analyze the behavior of the function when x becomes very large negative numbers. Since \(x^{2k}\) is a polynomial function with a positive even power, it will always be positive, and its value will increase as x becomes more negatively large. This means the limit as x approaches negative infinity for \(x^{2k}\) is infinity.

(c) Limit as x approaches infinity for \(x^{2k+1}\)

To find the limit as x approaches infinity for \(x^{2k+1}\), we need to analyze the behavior of the function when x becomes very large. Since \(x^{2k+1}\) is a polynomial function with a positive odd power, the value of the function will increase as x becomes larger. Therefore, the limit as x approaches infinity for \(x^{2k+1}\) is infinity.

(d) Limit as x approaches negative infinity for \(x^{2k+1}\)

To find the limit as x approaches negative infinity for \(x^{2k+1}\), we need to analyze the behavior of the function when x becomes very large negative numbers. Since \(x^{2k+1}\) is a polynomial function with a positive odd power, its value will decrease as x becomes more negatively large. This means the limit as x approaches negative infinity for \(x^{2k+1}\) is negative infinity.

(e) Limit as x approaches infinity for \(\sqrt{x} \sin x\)

To evaluate the limit as x approaches infinity for \(\sqrt{x} \sin x\), we need to analyze the behavior of the function when x becomes very large. The function consists of the product of a square root function and a sine function. The square root function increases as x becomes larger, but the sine function oscillates between -1 and 1. Since the sine function's oscillating behavior is not affected by x becoming large, the overall function also oscillates between smaller and larger magnitudes as x approaches infinity. Therefore, the limit does not exist for this function as x approaches infinity.

(f) Limit as x approaches infinity for \(e^{x}\)

To find the limit as x approaches infinity for the function \(e^{x}\), we need to analyze the behavior of the function when x becomes very large. Since \(e^{x}\) is an exponential function with a positive base, its value increases rapidly as x becomes larger. Therefore, the limit as x approaches infinity for \(e^{x}\) is infinity.

Key concepts

Infinite Limits

Infinite limits occur when a function increases or decreases without bound as its input grows larger or smaller. This is a common topic in real analysis, providing insight into the behavior of functions as they extend towards infinity.

A classical example is the polynomial function, particularly when x approaches infinity or negative infinity. In scenarios where the function increases without bound, we say the limit is infinity; conversely, if it decreases without bound, the limit is negative infinity.

• An example: The function \(x^{2k}\) approaches infinity as x goes to both positive and negative infinity, due to its even exponent ensuring positivity.
• For \(x^{2k+1}\), the limit depends on the direction: it becomes positive infinity if x approaches positive infinity, and negative infinity if x goes negative, thanks to the odd power.

Understanding infinite limits helps in grasping the long-term tendencies of functions, a crucial aspect for mathematical fields like calculus and beyond.

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables raised to the power of non-negative integers. In real analysis, they serve as essential examples for investigating limit behavior.

The key to understanding polynomial functions is their degree, which significantly influences their limit as x approaches large positive or negative values.

• For even-degree polynomials like \(x^{2k}\), the graph opens upwards for both positive and negative x, always tending towards positive infinity.
• Odd-degree polynomials such as \(x^{2k+1}\) behave differently. They rise to positive infinity in one direction (when x is positive) and fall to negative infinity in the other (when x is negative).

These characteristics make polynomial functions a fundamental component in learning about infinite limits, providing a clear structure to predict long-term behavior.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent, typically denoted as \(e^{x}\). They showcase distinct behavior compared to polynomials, especially concerning limits.

One of the most interesting features of exponential functions is their growth rate, which surpasses polynomial growth:

• As x approaches infinity, \(e^{x}\) grows rapidly towards infinity. This rapid acceleration makes exponential functions incredibly important in modeling real-world phenomena, like population dynamics or compound interest, where growth is multiplied rather than added.
• In contrast, any negative infinity limit for exponential functions appears untenable, given the positive constant base. But when an expression like \(e^{-x}\) is considered, it tends towards zero as x becomes very large.

Exponential functions thus illustrate how limits can describe the asymptotic behavior of functions diverging from polynomial norms.

Oscillatory Behavior

Oscillatory behavior in functions occurs when the output values swing between extremes rather than settling towards a single limit. The sine function \(\sin x\) is a prime example, directing oscillatory dynamics into analysis.

When evaluating limits for functions incorporating oscillation, such as \(\sqrt{x}\sin x\), their behavior can be quite complex:

• The \(\sin x\) component causes the function to oscillate between -1 and 1, regardless of x's size. This inherent fluctuation means there's no single value being approached.
• Adding the \(\sqrt{x}\) term complicates things further, as it increases without limit. The product of an oscillating term and a growing term results in unbounded oscillation, leading to a conclusion that the limit does not exist as x approaches infinity.

Oscillatory behavior is key in understanding why some limits fail to converge, offering insight into the dynamic nature of certain mathematical functions.