Question

In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow \infty}\left(x^{\alpha}-\log x\right) $$

Short answer

Question: Determine the limit \(\lim_{x \to \infty}(x^\alpha-\log{x})\), where \(f(x) = x^\alpha\) and \(g(x) = \log{x}\).

Step-by-step solution

Rewrite the limit

Rewrite the given limit in the form: $$\lim_{x \to \infty}(f(x)-g(x))$$, where \(f(x)=x^\alpha\) and \(g(x)=\log{x}\).

Explore the behavior of \(f(x)=x^\alpha\) as \(x \to \infty\)

Let's analyze the behavior of \(x^\alpha\) when \(x\) approaches infinity. If \(\alpha > 0\), then as \(x\) approaches infinity, \(x^\alpha\) will also approach infinity because any positive power of \(x\) will lead to an increase in the value. If \(\alpha < 0\), then as \(x \to \infty\), \(x^\alpha\) will approach 0, since raising a number to a negative power is equivalent to taking the reciprocal of the number raised to a positive power.

Explore the behavior of \(g(x)=\log{x}\) as \(x \to \infty\)

Now let's analyze the behavior of \(\log{x}\) when \(x\) approaches infinity. Since the logarithmic function has a positive growth rate, the logarithmic function will also approach infinity as \(x \to \infty\), but at a slower rate compared to \(x^\alpha\) when \(\alpha > 0\).

Compare the convergence of \(f(x)\) and \(g(x)\) as \(x \to \infty\)

Now we need to compare the convergence of \(f(x)\) and \(g(x)\) as \(x \to \infty\). When \(\alpha > 0\), \(f(x)\) converges to infinity at a faster rate than \(g(x)\). In that case, the limit will be dominated by \(f(x)\), and the limit would be infinity. When \(\alpha < 0,\) \(f(x)\) converges to 0, hence the limit will be dominated by \(g(x)\), and the value would be \(-\infty\).

Find the limit

Based on our analysis, the limit \(\lim_{x \to \infty}(x^\alpha-\log{x})\) can be defined as: If \(\alpha > 0\), then the limit is $$\lim_{x \to \infty}(x^\alpha-\log{x})=\infty$$ If \(\alpha < 0\), then the limit is $$\lim_{x \to \infty}(x^\alpha-\log{x})=-\infty$$

Key concepts

Infinity in calculus

In calculus, the concept of infinity is quite different from the usual understanding of numbers. It represents an idea of unboundedness rather than a specific value. When we say a variable approaches infinity, it means the variable is growing larger and larger without limit.
This concept is crucial in calculus because it helps us understand the behavior of functions as inputs grow very large or very small (approaching positive or negative infinity).
For example, in the problem "\(\lim _{x \rightarrow \infty}(x^{\alpha} - \log x)\)", the concept of infinity allows us to analyze the behavior of the function as \(x\) becomes extremely large. Analyzing such limits helps us determine how two functions (like \(x^{\alpha}\) and \(\log x\)) compare in their growth rates.

• Infinity signifies growth in a specific direction without bound.
• We use infinity to consider limits, which describe how functions behave as they stretch beyond the horizon of finite values.
• Infinity serves as a cornerstone for understanding the comparatives of functions in calculus.

By comprehending infinity, we're able to unlock deeper insights into how functions interact over vast domains.

Power functions

Power functions are functions of the form \(f(x) = x^{\alpha}\), where \(\alpha\) can be any real number. They play a major role in calculus due to their straightforward structure but diverse behaviors.
The exponent, \(\alpha\), determines how the function behaves as \(x\) becomes very large or very small.

• If \(\alpha > 0\), the function grows as \(x\) grows larger. This is why when \(x\rightarrow \infty\), \(x^{\alpha}\) approaches infinity too.
• If \(\alpha < 0\), the function shows opposite behavior. Here, \(x^{\alpha}\) approaches 0 instead of infinity because a negative exponent signifies the reciprocal of \(x^{\alpha}\).
• Special cases like \(\alpha = 0\) make the function a constant (1).
• For \(\alpha = 1\), it becomes a linear function \(f(x) = x\), growing directly with \(x\).

Understanding power functions helps appreciate their utility, as they are common in modeling scenarios across physics, engineering, and mathematics. In limits, comparing such functions' behavior gives insight into how rapidly they diverge to infinity.

Logarithmic functions

Logarithmic functions, usually expressed as \(g(x) = \log{x}\), describe how a number's magnitude increases slowly as its input grows larger. Unlike power functions, logarithms grow but at a sluggish pace.
As \(x\) approaches infinity, the logarithmic function also trends towards infinity, just not as swiftly. Here are some essential points about logarithmic functions:

• The base of a logarithm significantly affects its rate of growth, though base 10 (common logarithm) and base \(e\) (natural logarithm) are most prevalent.
• The slow growth makes logarithms useful in many fields, such as data science and complex computations, where exponential growth must be countered.
• In the exercise \(\lim_{x \to \infty}(x^{\alpha}-\log{x})\), the log function's role clarifies the expression's behavior as \(x\) increases.
• Logarithmic scales help manage open-ended ranges and visualize data effectively.

An understanding of logarithmic functions allows us to decipher scenarios where large-scale growth comparisons are needed and helps analyze dynamics where different types of growth occur simultaneously.