Math Problem Statement

kerjakann

Hitunglah nilai limit berikut ini: lim x→0 (sin 2x + x) / (2x + 3 sin 4x)

Solution

We are tasked with solving the following limit:

limx0sin(2x)+x2x+3sin(4x)\lim_{x \to 0} \frac{\sin(2x) + x}{2x + 3\sin(4x)}

Step-by-Step Solution:

  1. Substitute x=0x = 0 into the expression directly: sin(2(0))+02(0)+3sin(4(0))=0+00+3(0)=00\frac{\sin(2(0)) + 0}{2(0) + 3\sin(4(0))} = \frac{0 + 0}{0 + 3(0)} = \frac{0}{0} Since this results in an indeterminate form 00\frac{0}{0}, we can apply L'Hopital's Rule.

  2. Differentiate the numerator and the denominator:

    • The numerator: f(x)=sin(2x)+xf(x) = \sin(2x) + x [ f'(x) = \frac{d}{dx}[\sin(2x)] + \frac{d}{dx}[x] = 2\cos(2x) + 1 ]
    • The denominator: g(x)=2x+3sin(4x)g(x) = 2x + 3\sin(4x) [ g'(x) = \frac{d}{dx}[2x] + \frac{d}{dx}[3\sin(4x)] = 2 + 12\cos(4x) ]
  3. Evaluate the limit using the derivatives: limx0f(x)g(x)=limx02cos(2x)+12+12cos(4x)\lim_{x \to 0} \frac{f'(x)}{g'(x)} = \lim_{x \to 0} \frac{2\cos(2x) + 1}{2 + 12\cos(4x)} Substituting x=0x = 0 into this expression: 2cos(0)+12+12cos(0)=2(1)+12+12(1)=314\frac{2\cos(0) + 1}{2 + 12\cos(0)} = \frac{2(1) + 1}{2 + 12(1)} = \frac{3}{14}

Thus, the value of the limit is:

314\boxed{\frac{3}{14}}

Would you like further details on any step or have any other questions?

Relative questions:

  1. How does L'Hopital's rule work for indeterminate forms like 00\frac{0}{0}?
  2. What are some common indeterminate forms in calculus, and how are they resolved?
  3. Can the Squeeze Theorem be applied to this problem instead of L'Hopital's Rule?
  4. Why does direct substitution fail for some limits, leading to indeterminate forms?
  5. How do trigonometric limits like limx0sinxx\lim_{x \to 0} \frac{\sin x}{x} 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.

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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