Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (tan 7x) / (sin x)

Verified step by step guidance

1. Theorem 3.10 refers to the limit of a function as it approaches a point, often involving L'Hôpital's Rule when dealing with indeterminate forms like 0/0. First, check if the limit results in an indeterminate form by substituting x = 0 into the expression (tan 7x) / (sin x).
2. Substitute x = 0 into tan(7x) and sin(x). Both tan(7x) and sin(x) approach 0 as x approaches 0, resulting in the indeterminate form 0/0. This indicates that L'Hôpital's Rule can be applied.
3. Apply L'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then the limit can be found by differentiating the numerator and the denominator separately. Differentiate the numerator tan(7x) and the denominator sin(x) with respect to x.
4. The derivative of tan(7x) with respect to x is 7sec²(7x), and the derivative of sin(x) with respect to x is cos(x). Substitute these derivatives back into the limit expression, resulting in lim x→0 (7sec²(7x)) / (cos(x)).
5. Evaluate the new limit expression as x approaches 0. Substitute x = 0 into the expression 7sec²(7x) / cos(x). Since sec(7x) = 1/cos(7x), and cos(0) = 1, the expression simplifies to 7 * (1/1)² / 1, which can be further simplified to find the limit.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for evaluating expressions that may be indeterminate, such as 0/0. In this context, we need to analyze the behavior of the function as x approaches 0.

Theorem 3.10 (L'Hôpital's Rule)

Theorem 3.10, commonly known as L'Hôpital's Rule, provides a method for evaluating limits of indeterminate forms like 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, then re-evaluating the limit.

Trigonometric Limits

Trigonometric limits involve the behavior of trigonometric functions as their arguments approach specific values. For example, limits involving sin(x) and tan(x) as x approaches 0 are particularly important, as they often simplify to well-known values, such as sin(x)/x approaching 1. Recognizing these standard limits can greatly aid in solving more complex limit problems.