Problem 67 - Precalculus

Question

For Exercises 65 through 70 , evaluate each limit. $$\lim _{x \rightarrow \infty} \frac{\sqrt{12 x^{2}-6 x+1}}{7 x}$$

Step-by-Step Solution

Answer

The limit is \( \frac{2\sqrt{3}}{7} \).

Step 1: Identify the Dominant Terms

As we are evaluating the limit as \(x\) approaches infinity, we need to identify the dominant terms in both the numerator and the denominator. In the expression \(\sqrt{12x^2 - 6x + 1}\), the dominant term is \(\sqrt{12x^2}\). In the denominator, the dominant term is \(7x\).

Step 2: Simplify the Expression

Considering only the dominant terms, simplify the fraction: \[ \frac{\sqrt{12x^2}}{7x} = \frac{\sqrt{12}\cdot x}{7x}.\] Here, \(x\) can be canceled out from the numerator and the denominator, giving us \(\frac{\sqrt{12}}{7}\).

Step 3: Evaluate the Simplified Limit

After canceling out \(x\), we obtain \(\frac{\sqrt{12}}{7}\). This is a constant value. Thus, the limit simplifies to a constant value, which is \(\frac{\sqrt{12}}{7}\).

Step 4: Final Calculation

To express \(\sqrt{12}\) in its simplest form, note that \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\). Thus, the limit evaluates to \[ \frac{2\sqrt{3}}{7}.\]

Key Concepts

Dominant Terms

When working with calculus limits, especially as variables approach infinity, identifying dominant terms is crucial. Dominant terms are the parts of the expression that have the greatest impact, as their magnitude grows significantly faster than other terms. This happens in both the numerator and the denominator.
Consider the expression \( \sqrt{12x^2 - 6x + 1} \). Here, \( 12x^2 \) is the dominant term since it's the part of the expression that grows the fastest as \( x \) becomes large. Comparatively, \( 6x \) and \( 1 \) become negligible.

• Focus on dominant terms for simplification: In numerator or under square roots.
• Discard lesser terms when \( x \to \infty \).

Identifying these terms allows us to simplify complex expressions, which is essential for evaluating limits, where smaller terms become negligible.

Simplifying Algebraic Expressions

Once you've identified the dominant terms in an expression, the next step involves simplifying the algebraic expression. This often means setting aside smaller, less significant terms.
For the limit problem given: \( \lim _{x \rightarrow \infty} \frac{\sqrt{12 x^{2}}}{7 x} \), we notice that \( x \) can be factored out from both the numerator and the denominator.

• Divide both the top and bottom by \( x \), which cancels out.
• Focus on simplifying the dominant term expression.

Thus, the expression simplifies to \( \frac{\sqrt{12} \cdot x}{7x} \), and the x's cancel, resulting in \( \frac{\sqrt{12}}{7} \). The absence of \( x \) indicates the behavior of the limit as \( x \) goes to infinity. These steps are pivotal in evaluating limits efficiently.

Radical Expressions

Dealing with radical expressions, like roots and square roots, often requires extra steps to make them simpler. Simplifying radicals involves breaking them down into products of simpler numbers.
For instance, \( \sqrt{12} \) can be broken down as follows: - Recognize that \( 12 = 4 \times 3 \).- Since \( 4 \) is a perfect square (as \( 4 = 2^2 \)), it can be extracted out of the radical.
Thus, \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \).
When you simplify radicals, you reduce them to their simplest form, which is often needed for the clean presentation of your final answer, especially when working within limits. This simplification to \( \frac{2\sqrt{3}}{7} \) presents the limit in its most reduced form.