Pembuktian Rumus Limit Fungsi Trigonometri (sinx/x)

Introduction

This tutorial will guide you through the process of proving the limit of the trigonometric function sin(x)/x as x approaches 0. Understanding this limit is fundamental in calculus, particularly in the study of derivatives and integrals of trigonometric functions. It serves as a building block for more complex mathematical theories and applications.

Step 1: Understand the Limit Concept

• Begin by grasping what it means to find the limit of a function as it approaches a specific value.
• The limit of sin(x)/x as x approaches 0 is a classic example in calculus that often requires a geometric interpretation and algebraic manipulation.

Step 2: Visualize the Function

• Draw the unit circle and locate the angle x (in radians).
• Identify the following components:
  • The length of the arc corresponding to angle x (which is x).
  • The vertical line segment representing sin(x).
  • The tangent line at the point (1,0) which intersects the vertical line.

This visualization helps to understand why the limit approaches 1.

Step 3: Establish the Squeeze Theorem

• Use the Squeeze Theorem to prove the limit:
  • Show that for x in the interval (0, π/2), the following inequalities hold:
    • sin(x) < x < tan(x)
  • Recognize that tan(x) can be expressed as sin(x)/cos(x).

From these inequalities, you can derive the following:

• sin(x)/x < 1 < 1/cos(x)

Step 4: Take the Limit

• As x approaches 0, apply the inequalities:
  • The limit of sin(x)/x as x approaches 0 is squeezed between two bounds converging to 1.

Formally, you can express this as:

• lim (x -> 0) sin(x)/x = 1

Step 5: Verify with the Derivative

• You can also verify this limit using L'Hôpital's Rule, which applies to indeterminate forms like 0/0.
• Differentiate the numerator and denominator:
  • The derivative of sin(x) is cos(x).
  • The derivative of x is 1.

Thus, applying L'Hôpital's Rule gives:

lim (x -> 0) sin(x)/x = lim (x -> 0) cos(x)/1 = cos(0) = 1

Conclusion

In this tutorial, you learned how to prove that lim (x -> 0) sin(x)/x = 1 using geometric visualization, the Squeeze Theorem, and L'Hôpital's Rule. Understanding this limit is essential in calculus and has numerous applications in higher mathematics. As a next step, practice applying these concepts to other trigonometric limits and explore their implications in calculus further.