Problem 4 - Calculus with Concepts in Calculus

Question

Find the given limit. $$ \lim _{x \rightarrow \infty} \frac{4 x^{2}}{\sqrt{2} x-3} $$

Step-by-Step Solution

Answer

The limit is infinity.

Step 1: Understand the Expression

We have the limit \( \lim _{x \rightarrow \infty} \frac{4 x^{2}}{\sqrt{2} x-3} \). We want to determine what happens to this expression as \( x \) gets very large.

Step 2: Simplify the Expression

Divide both the numerator and the denominator by the highest power of \( x \) found in the denominator, which in this case is \( x \). The expression becomes: \[\lim _{x \rightarrow \infty} \frac{4x^2/x}{\sqrt{2} - 3/x} = \lim _{x \rightarrow \infty} \frac{4x}{\sqrt{2} - 3/x}.\]

Step 3: Evaluate the Simplified Expression

As \( x \rightarrow \infty \), the term \(-3/x\) in the denominator goes to zero. This simplifies the limit to:\[\lim _{x \rightarrow \infty} \frac{4x}{\sqrt{2}}.\]As \( x \rightarrow \infty \), \( 4x \) also goes to infinity, implying that the entire expression goes to infinity.

Step 4: Conclusion

Since the expression simplifies to \( \lim _{x \rightarrow \infty} \frac{4x}{\sqrt{2}} \), and as \( x \rightarrow \infty \), \( 4x \rightarrow \infty \), the limit is infinity.

Key Concepts

Asymptotic Behavior

The term "asymptotic behavior" describes how a function behaves as its input approaches a particular value or heads towards infinity. In our exercise, we examine the asymptotic behavior of the function \[ \frac{4x^2}{\sqrt{2}x - 3} \]as \( x \) grows larger and larger.
When looking at functions at infinity, it's essential to simplify the expression to identify which component tends to dominate.

• Asymptotic analysis helps predict if a function will approach a finite number, continue to grow indefinitely, or level off as a horizontal asymptote.
• A function can either approach a particular value (asymptote) or diverge to some form of infinity.

In our specific example, as \( x \) becomes very large, the quadratic term \( 4x^{2} \) in the numerator grows significantly faster than the linear term \( \sqrt{2}x - 3 \) in the denominator, leading the function toward infinity. This behavior underscores that the leading terms' degrees primarily dictate the asymptotic behavior of polynomial ratios.

Indeterminate Forms

Indeterminate forms often appear when evaluating limits and need careful treatment to avoid incorrect conclusions. Typical indeterminate forms are \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), and \(0 \cdot \infty\), among others. In our problem, the initial setup doesn't result in an indeterminate form directly, but if approached improperly, one could mistakenly assume it as such.
When we simplify the given limit: \[ \lim_{x \to \infty} \frac{4x}{\sqrt{2} - \frac{3}{x}} \]the disappearance of the \(-\frac{3}{x}\) term as \( x \to \infty \) reveals why it's essential not to hastily conclude before simplifying.

• In calculus, techniques like L'Hôpital's rule or algebraic manipulation can resolve true indeterminate forms.
• Simplifying expressions by factoring or rationalizing helps in eliminating these forms to see the true limit behavior.

This careful analysis prevents misinterpretation and solidifies understanding of how limits act around infinity or zero.

Infinity in Calculus

Infinity in calculus is a concept that deals with unbounded growth or decrease of functions and how they behave in extreme cases. In our problem, we use infinity to determine the behavior of\[ \lim _{x \rightarrow \infty} \frac{4 x^{2}}{\sqrt{2} x-3}. \]The idea of infinity helps students understand:

• How a function behaves when inputs grow indefinitely large.
• The implications of terms that vanish (like \(-3/x\) in our limit) and those that dominate.
• How to handle limits resulting in infinities by identifying trends in the growth of functions.

In the context of our specific limit, since the leading term in the numerator, \(4x^2\), grows faster than any linear term in the denominator, it dictates that the function increases towards infinity.
The limit's solution reflects that asymptotic and indeterminate form analyses lead to determining how functions behave in calculus when they rarely "settle" into a simple value but instead grow without bounds.