Math Problem Statement
kerjakann
Solution
We are tasked with solving the following limit:
Step-by-Step Solution:
-
Substitute into the expression directly: Since this results in an indeterminate form , we can apply L'Hopital's Rule.
-
Differentiate the numerator and the denominator:
- The numerator: [ f'(x) = \frac{d}{dx}[\sin(2x)] + \frac{d}{dx}[x] = 2\cos(2x) + 1 ]
- The denominator: [ g'(x) = \frac{d}{dx}[2x] + \frac{d}{dx}[3\sin(4x)] = 2 + 12\cos(4x) ]
-
Evaluate the limit using the derivatives: Substituting into this expression:
Thus, the value of the limit is:
Would you like further details on any step or have any other questions?
Relative questions:
- How does L'Hopital's rule work for indeterminate forms like ?
- What are some common indeterminate forms in calculus, and how are they resolved?
- Can the Squeeze Theorem be applied to this problem instead of L'Hopital's Rule?
- Why does direct substitution fail for some limits, leading to indeterminate forms?
- How do trigonometric limits like play a role in solving limits?
Tip:
When using L'Hopital's Rule, always check if the limit is initially in an indeterminate form, as the rule applies only under those conditions.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Limits
Trigonometric Functions
Calculus
Formulas
L'Hopital's Rule
Derivative of sin(x) and cos(x)
Basic limit properties
Theorems
L'Hopital's Rule
Suitable Grade Level
Grades 11-12 or introductory college-level calculus
Related Recommendation
Limit of (6x - sin(2x)) / (2x + 3sin(4x)) as x approaches 0
Limit of (sin(x) - x) / (4x^3) using L'Hôpital's Rule
Limit of sin(3x)/2x as x approaches 0 using L'Hopital's Rule
Limit Problem Involving L'Hôpital's Rule and Trigonometric Functions
Evaluate Limit as x Approaches 0 of (x * sin 3x) / (1 - cos 4x) using L'Hôpital's Rule