Question

Begin by plotting the function in an appropriate window. $$ \lim _{x \rightarrow-\infty}\left(\sqrt{2 x^{2}+3 x}-\sqrt{2 x^{2}-5}\right) $$

Short answer

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

Step-by-step solution

Identify the Expression

The function provided is \[ \lim _{x \rightarrow-\infty}\left(\sqrt{2 x^{2}+3 x}-\sqrt{2 x^{2}-5}\right) \]We need to understand that this limit expression involves two square root terms.

Simplify the Expression

To simplify the limit, we rationalize the expression by multiplying the numerator and denominator by the conjugate:\[ \lim _{x \rightarrow-\infty} \frac{\left(\sqrt{2x^2+3x} - \sqrt{2x^2-5}\right) \cdot \left(\sqrt{2x^2+3x} + \sqrt{2x^2-5}\right)}{\left(\sqrt{2x^2+3x} + \sqrt{2x^2-5}\right)} \]This turns the numerator into a difference of squares.

Expand and Simplify the Numerator

Using the identity \((a-b)(a+b) = a^2 - b^2\), we have:\[ \left(\sqrt{2x^2+3x}\right)^2 - \left(\sqrt{2x^2-5}\right)^2 = (2x^2+3x) - (2x^2-5) \]Simplifying this gives:\[ 3x + 5 \]

Analyze the Denominator at Negative Infinity

Since both square roots tend to infinity, for large negative \(x\), the dominant term in each root is \(\sqrt{2x^2}\) which is approximately \(|\sqrt{2}|x|\). However, as \(x \rightarrow -\infty\), \(|x| = -x\), so it behaves like \[ \sqrt{2}|x| = \sqrt{2}(-x) \].

Simplify the Expression Further

The fraction before the limit becomes:\[ \frac{3x + 5}{\sqrt{2}|x| + \sqrt{2}|x|} = \frac{3x + 5}{2\sqrt{2}|x|} \]As \(x \rightarrow -\infty\), the \(x\) term dominates in both the numerator and the denominator.

Evaluate the Limit

Since \(3x/2\sqrt{2}|x| = 3/2\sqrt{2}\) as \(x\) approaches \(-\infty\), and the \(5/{2\sqrt{2}|x|}\) term tends to zero, the limit becomes:\[ \lim_{x \rightarrow-\infty} \frac{3x}{2\sqrt{2}|x|} + \lim_{x \rightarrow-\infty} \frac{5}{2\sqrt{2}|x|} = \frac{3}{2\sqrt{2}} + 0 \]

Conclusion

Thus, the limit evaluates to a constant value, which means:\[ \lim _{x \rightarrow-\infty}\left(\sqrt{2 x^{2}+3 x}-\sqrt{2 x^{2}-5}\right) = \frac{3}{2\sqrt{2}} \]

Key concepts

Limit evaluation

When faced with the limit evaluation of a function, the goal is to determine the behavior of the function as the variable approaches a particular value, which can be either finite or infinite. In calculus, understanding limits is essential for grasping more complex concepts like derivatives and integrals. Limit evaluation helps us predict the behavior of functions that become undefined at certain points or approach infinity. For the given problem, the task is to find the limit as \(x\) approaches \(-\infty\). This means we are interested in the long-term behavior of the function as \(x\) becomes very large in the negative direction, effectively analyzing how the function behaves at extreme values. Evaluating limits often involves algebraic manipulation or applying limit laws, such as direct substitution, factoring, or rationalization, to simplify the expression, making it easier to analyze.

Rationalization technique

Rationalization is a technique used to simplify expressions involving square roots or other radicals. It is especially helpful in limit evaluation when the direct substitution results in indeterminate forms like \(\frac{0}{0}\). By multiplying the numerator and denominator by the conjugate, we can transform the expression into a form that is easier to manage.

• Start by identifying the radical terms that need rationalization.
• Multiply the expression by its conjugate. The conjugate of a term \((a - b)\) is \((a + b)\).
• This process results in a difference of squares, which simplifies radicals by eliminating them through squaring.

In our exercise, we used rationalization to turn the expression involving square roots into a simpler quadratic expression. This assists in unraveling complex roots, allowing us to focus on the leading terms that dominate at infinity.

Square root functions

Square root functions are a specific type of algebraic function. The principal square root function, commonly represented as \(\sqrt{x}\), extracts the non-negative root of a number. In expressions involving square roots, especially those within limits, the radical's complexity requires careful consideration when simplifying.Key characteristics of square root functions:

• Non-linear, curving nature of graphs.
• As \(x\rightarrow 0\), square roots yield values close to zero, but are undefined for negative numbers under standard real number operations.
• At infinity, square roots grow slower compared to linear functions but dominate constant terms.

In the problem, both square root terms played a critical role in constructing the limit expression. When calculating the limit, we focus on the highest-degree terms \(x^2\) in the square root, since they greatly determine the asymptotic behavior as \(x\) approaches \(-\infty\).

Asymptotic behavior

Asymptotic behavior refers to how a function behaves as its variable approaches infinity or a specific point. It provides insights into the function's growth or decay rate and identifies trends that are not visible at smaller scales or nearby values. This concept is crucial for understanding the long-term characteristics of a function. In our limit problem, observing asymptotic behavior helps determine how the function described by the limit behaves as \(x\) grows very large in negative terms.

• For limit expressions with radicals, focus on terms with the highest degree.
• Understand that dominant terms will heavily influence the ultimate value of the limit.
• Expressions with square roots simplify to involve linear-like terms when \(x\) approaches positive or negative infinity.

By understanding the asymptotic behavior, we can predict that the function behaves like a constant as \(x\) approaches \(-\infty\), enabling us to approximate and ultimately evaluate the limit correctly.