17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→0 (sin x - x) / 7x³

Verified step by step guidance

1. First, identify the form of the limit as x approaches 0. Substitute x = 0 into the expression (sin x - x) / 7x³ to see if it results in an indeterminate form like 0/0.
2. Since substituting x = 0 gives 0/0, l'Hôpital's Rule is applicable. This rule states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.
3. Apply l'Hôpital's Rule by differentiating the numerator and the denominator. The derivative of the numerator, sin x - x, is cos x - 1. The derivative of the denominator, 7x³, is 21x².
4. Re-evaluate the limit using the derivatives: lim_x→0 (cos x - 1) / 21x². Substitute x = 0 again to check if the expression is still indeterminate.
5. If the expression is still indeterminate, apply l'Hôpital's Rule again. Differentiate the numerator and the denominator once more: the derivative of cos x - 1 is -sin x, and the derivative of 21x² is 42x. Evaluate the limit of (-sin x) / 42x as x approaches 0.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 0 helps determine the behavior of the function (sin x - x) / 7x³ near that point.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits, especially when direct substitution is not possible.

Taylor Series Expansion

The Taylor Series Expansion is a way to express functions as infinite sums of terms calculated from the values of their derivatives at a single point. For small values of x, the sine function can be approximated using its Taylor series, which helps in simplifying expressions like sin x - x. This concept is particularly useful in limit problems where direct evaluation leads to indeterminate forms.