Math Problem Statement

Nilai limit lim(x→0) dari (4sin^2(2x))/(xtan(2x)) dan lim(x→π/4) dari (sin(x) - cos(x))/(tan(x) - 1)

Solution

Let's address the two limit problems visible in the image.

Problem 5:

limx04sin2(2x)xtan(2x)\lim_{x \to 0} \frac{4 \sin^2 (2x)}{x \tan (2x)}

Solution:

To solve this, we will break it down step by step.

  1. Use the small-angle approximations for sine and tangent when x0x \to 0:

    • sin(2x)2x\sin(2x) \approx 2x
    • tan(2x)2x\tan(2x) \approx 2x
  2. Now, substitute these approximations into the expression: [

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometry
Indeterminate Forms

Formulas

sin(x) ≈ x for small x
tan(x) ≈ x for small x
L'Hopital's Rule

Theorems

L'Hopital's Rule
Small Angle Approximation

Suitable Grade Level

Grades 11-12 (Pre-Calculus or Calculus)