Problem 80 - Thomas Calculus

Question

For what values of \(a\) and \(b\) is $$\lim _{x \rightarrow 0}\left(\frac{\tan 2 x}{x^{3}}+\frac{a}{x^{2}}+\frac{\sin b x}{x}\right)=0 ?$$

Step-by-Step Solution

Answer

The values are \(a = -2\) and \(b = 0\).

Step 1: Analyze the Limit Expression

We need to find the limit \(\lim_{x \rightarrow 0}\left(\frac{\tan 2x}{x^{3}}+\frac{a}{x^{2}}+\frac{\sin bx}{x}\right)\) and ensure it equals zero. The limit expression consists of three separate terms, each potentially contributing different aspects depending on the values of \(a\) and \(b\).

Step 2: Simplify the Trigonometric Terms

For the first term, as \(x\to 0\), we can approximate \(\tan 2x \approx 2x\), so \(\frac{\tan 2x}{x^3} \approx \frac{2x}{x^3} = \frac{2}{x^2}\). For the third term, \(\sin bx \approx bx\), leading to \(\frac{\sin bx}{x} \approx b\). Simplifying using these approximations, we rewrite the expression: \[\frac{2}{x^2} + \frac{a}{x^2} + b.\]

Step 3: Combine and Set Limit to Zero

The limit requires that when \(x\to 0\), the expression \(\frac{2 + a}{x^2} + b\) must equal zero. Since \(\frac{2 + a}{x^2}\) diverges unless \(2 + a = 0\), we must have \(a = -2\) for the divergent part to be eliminated. Additionally, ensuring \(b = 0\) will prevent the constant part from leaving a non-zero value.

Step 4: Verify the Values of \(a\) and \(b\)

By setting \(a = -2\) and \(b = 0\), we have verified that the expression becomes zero when \(x\to 0\). The relevant limit simplifies to zero as: \[ \lim_{x \to 0} \left(\frac{\tan 2x}{x^3} - \frac{2}{x^2} + 0\right) = 0,\] confirming the values of \(a\) and \(b\).

Key Concepts

Trigonometric Limits

In calculus, working with trigonometric limits is crucial, especially when dealing with expressions containing sine, cosine, and tangent. Trigonometric functions often behave uniquely near certain points, such as zero. For small values of the angle, approximations like \(\tan x \approx x\) and \(\sin x \approx x\) simplify calculations. This is because, as angles become extremely small, the ratio of side lengths (defining sine and tangent) closely resembles the angle in radians itself.
In the problem at hand, understanding trigonometric limits helps us to simplify \(\tan 2x\) near zero to 2x. By making use of these approximations, complex expressions become more manageable, facilitating easier calculation of limits. Remember, these approximations are valid only when x is approaching zero, making them especially useful in limit calculus.

Limit Approximation

Limit approximation is the technique of simplifying a function by replacing it with an easier-to-evaluate expression. This is particularly useful when dealing with functions that become complex as x approaches a certain value.
In our problem, we approximate two trigonometric expressions: \(\tan 2x\) is approximated to \(2x\) and \(\sin bx\) is approximated to \(bx\). These approximations lead to simpler terms \(\frac{2}{x^2}\) and \(b\) in the expression.
Such approximations work under the assumption that x is infinitesimally small, allowing us to focus on leading terms that dominate behavior at these limits. This approach can simplify evaluating whether an expression tends to zero or diverges, thereby providing crucial insights into the original problem.

Limit Verification

Verifying a limit ensures that the calculated approximations and results are correct. In our exercise, limit verification involves confirming that the chosen values of a and b satisfy the condition set by the limit.
The expression \[\frac{\tan 2x}{x^3} + \frac{a}{x^2} + \frac{\sin bx}{x} = \frac{2 + a}{x^2} + b\]must equal zero as x approaches zero.

• The term \(\frac{2 + a}{x^2}\) tends to diverge unless \(2 + a = 0\), ensuring stability in the expression. Hence, \(a = -2\) eliminates this divergence.
• The constant term \(b\) should also be zero to ensure the entire expression tends to zero.

By setting \(a = -2\) and \(b = 0\), we achieve the limit condition, confirming the solution is accurate.

Continuous Functions

Continuous functions are pivotal in analyzing limits, as they ensure that small changes in x lead to small changes in the function value. A function is continuous at a point if it aligns perfectly with its limit at that point.
In our problem, continuity of each component is crucial at the point x=0 for which we are analyzing the limit.

• As x approaches zero, the function formed by \(\tan 2x\) should continue smoothly, which is aided by its approximation to \(2x\).
• Likewise, \(\sin bx\) should behave continuously, approximating to \(bx\).

By ensuring that both \(a\) and \(b\) are chosen so the overall expression remains stable and zero as x approaches 0, we align the limit with this essential property of continuous functions. This approach secures a reliable foundation for calculating limits involving trigonometric and polynomial terms.