Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{x \rightarrow \infty} \frac{3 \sqrt{x^{3}}+3 x}{\sqrt{2 x^{3}}} $$

Step-by-Step Solution

Answer

The limit is \( \frac{3\sqrt{2}}{2} \).

Step 1: Simplify the Expression

We start with the expression \( \lim_{x \rightarrow \infty} \frac{3 \sqrt{x^3} + 3x}{\sqrt{2x^3}} \). Notice that \( \sqrt{x^3} = x^{3/2} \). Therefore, the expression under the square root in the denominator becomes \( \sqrt{2}x^{3/2} \). Consequently, the expression becomes \( \lim_{x \rightarrow \infty} \frac{3x^{3/2} + 3x}{\sqrt{2}x^{3/2}} \).

Step 2: Divide by the Highest Power of x

Identify the highest power of \( x \) in the expression. Here, it is the \( x^{3/2} \) from the first term of the numerator. Divide every term in the numerator and the denominator by \( x^{3/2} \):\[ \lim_{x \rightarrow \infty} \frac{\frac{3x^{3/2}}{x^{3/2}} + \frac{3x}{x^{3/2}}}{\frac{\sqrt{2}x^{3/2}}{x^{3/2}}} = \lim_{x \rightarrow \infty} \frac{3 + \frac{3}{x^{1/2}}}{\sqrt{2}} \].

Step 3: Evaluate the Limit as x Approaches Infinity

Simplify the expression by evaluating the limit for each term as \( x \rightarrow \infty \). The term \( \frac{3}{x^{1/2}} \) tends to zero since the denominator becomes very large. Therefore, the numerator approaches \( 3 + 0 = 3 \):\[ \lim_{x \rightarrow \infty} \frac{3 + \frac{3}{x^{1/2}}}{\sqrt{2}} = \frac{3}{\sqrt{2}} \].

Step 4: Rationalize the Denominator (Optional)

If needed for simplification, you can rationalize the denominator: multiply the numerator and the denominator by \( \sqrt{2} \) to get:\[ \frac{3\sqrt{2}}{2} \].

Key Concepts

Infinity in Limits

Understanding limits as a variable approaches infinity is crucial for solving calculus problems. **Infinity** is not a number, but rather a concept that describes something that grows indefinitely large. In the context of limits, it helps to determine the behavior of functions as they expand. For instance, as in our original exercise, when the variable \( x \) tends toward infinity, we observe the dominant terms in the expression, which typically have the highest powers of \( x \).

When evaluating limits at infinity:

• Focus on the leading terms, as they influence the end behavior.
• Lesser powers become negligible, essentially reducing complexity.
• Ultimately, you assess how the function behaves as \( x \) expands without bound.

Applying these concepts maintains accuracy while simplifying the computational process.

Rationalization Techniques

Rationalization is a significant technique in simplifying expressions, especially when involving roots. This process makes calculations cleaner and often easier to interpret. For instance, during the final step of our exercise, we had an optional rationalization of the denominator.

The process involves:

• Multiplying both the numerator and the denominator by the conjugate of the denominator.
• This aligns terms such that roots in the denominator are eliminated, leading to a simplified form.
• Helps when precise numerical approximation is needed.

While rationalization is not always necessary, it often provides a neater solution. It's especially useful when dealing with complex expressions.

Simplifying Expressions in Limits

Simplifying expressions is a cornerstone of solving limits. It transforms complex functions into simpler ones, facilitating easier comprehension and calculation.In our example, simplifying involves dividing each term by the highest power of \( x \). This step is vital because:

• It reduces the expression to its most essential terms.
• Non-dominant terms become increasingly insignificant as \( x \) grows larger.
• The process ensures you focus only on the significant contributing terms.

Through simplification, the cubed root elements, fractions, and other components are streamlined, revealing the core behavior of the expression.Mastering this technique enhances both speed and accuracy in solving limit problems.