Problem 27 - Calculus Early Transcendentals

Question

15-36 Find the limit. $$ \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+a x}-\sqrt{x^{2}+b x}\right) $$

Step-by-Step Solution

Answer

The limit is \( \frac{a-b}{2} \).

Step 1: Identify the Expression

We need to find the limit of the expression \( \lim_{x \to \infty} \left( \sqrt{x^2 + ax} - \sqrt{x^2 + bx} \right) \). As \( x \to \infty \), the terms \( ax \) and \( bx \) become insignificant compared to \( x^2 \).

Step 2: Simplify the Expression

To simplify, multiply and divide the expression by the conjugate: \( \frac{\left(\sqrt{x^2 + ax} - \sqrt{x^2 + bx}\right) \times \left(\sqrt{x^2 + ax} + \sqrt{x^2 + bx}\right)}{\sqrt{x^2 + ax} + \sqrt{x^2 + bx}} \). This results in \( \frac{(ax - bx)}{\sqrt{x^2 + ax} + \sqrt{x^2 + bx}} \).

Step 3: Further Simplify

The expression now is \( \frac{(a-b)x}{\sqrt{x^2 + ax} + \sqrt{x^2 + bx}} \). Factor \( x \) out of the square roots: \( \sqrt{x^2 + ax} = x\sqrt{1 + \frac{a}{x}} \) and \( \sqrt{x^2 + bx} = x\sqrt{1 + \frac{b}{x}} \).

Step 4: Limit Calculation

Substitute the simplified roots back: \[ \frac{(a-b)x}{x\sqrt{1 + \frac{a}{x}} + x\sqrt{1 + \frac{b}{x}}} = \frac{(a-b)}{\sqrt{1 + \frac{a}{x}} + \sqrt{1 + \frac{b}{x}}} \]. As \( x \to \infty \), \( \sqrt{1 + \frac{a}{x}} \to 1 \) and \( \sqrt{1 + \frac{b}{x}} \to 1 \).

Step 5: Evaluate the Final Limit

Substituting these limits in, we have \[ \frac{(a-b)}{1 + 1} = \frac{a-b}{2} \]. Therefore, the limit is \( \frac{a-b}{2} \).

Key Concepts

Conjugate Multiplication

Conjugate multiplication is a powerful algebraic technique used to simplify expressions, particularly those involving square roots. By multiplying by the conjugate, we can eliminate the radicals, which often makes the expression easier to handle. In our original exercise, the expression contains two square roots: \( \sqrt{x^2 + ax} \) and \( \sqrt{x^2 + bx} \).

• The conjugate of an expression \( a - b \) is \( a + b \).
• Multiplying an expression by its conjugate turns the difference of two squares into a simpler form without square roots.

Example:We start with the expression \( \sqrt{x^2 + ax} - \sqrt{x^2 + bx} \). Multiplying this by its conjugate \( \sqrt{x^2 + ax} + \sqrt{x^2 + bx} \), results in:
\[ (\sqrt{x^2 + ax})^2 - (\sqrt{x^2 + bx})^2 = (x^2 + ax) - (x^2 + bx) = ax - bx \]
This step simplifies the expression under the limit. It helps us to focus solely on the linear terms (\( ax \) and \( bx \)) since the quadratic components cancel out.

Infinity Limit

An infinity limit involves finding the value that a function approaches as the input grows larger and larger towards infinity. When working with infinity limits, especially in calculus, it's common to encounter forms where expressions become very large. We often focus on the leading terms since they dominate the behavior of the function as \( x \) increases.

• The highest power of \( x \) usually dictates the growth of the function.
• For functions where \( x \to \infty \), terms with lower powers of \( x \) (or constants) become less significant.

Application:In our expression \( \lim_{x \to \infty} \left( \sqrt{x^2 + ax} - \sqrt{x^2 + bx} \right) \), the terms \( ax \) and \( bx \) become less significant compared to \( x^2 \). By focusing on the limit behavior of the dominating terms, and simplifying through conjugate multiplication, we get a clearer picture of the behavior as \( x \) approaches infinity. Eventually, this helped us reduce and find that the limit equals \( \frac{a-b}{2} \).

Square Root Simplification

Square root simplification is necessary for solving limits involving radicals, by making expressions easier to evaluate, especially when approaching infinity. Simplifying square roots often involves factoring out the highest power of \( x \), which clarifies the behavior of the expression at the limits.

• Factoring helps handle radicals that approach constants as \( x \to \infty \).
• Important when expressions contain variables with significant differences in exponents.

Example:To simplify \( \sqrt{x^2 + ax} \), we notice that, for large \( x \), \( x^2 \) dominates. Factoring \( x^2 \) gives:
\[ \sqrt{x^2(1 + \frac{a}{x})} = x\sqrt{1 + \frac{a}{x}} \]Similarly for \( \sqrt{x^2 + bx} \):
\[ \sqrt{x^2(b + x)} = x\sqrt{1 + \frac{b}{x}} \]As \( x \to \infty \), \( \frac{a}{x} \to 0 \) and \( \frac{b}{x} \to 0 \), simplifying each radical further to approximately 1. This simplification thus helps us evaluate the limit correctly to \( \frac{a-b}{2} \).