Limit Fungsi Aljabar : Metode Subtitusi Langsung dan Pemfaktoran

Introduction

This tutorial will guide you through the process of finding limits in algebra using the direct substitution method and factoring. Understanding these techniques is crucial for solving calculus problems effectively, particularly when dealing with indeterminate forms like 0/0.

Step 1: Understanding Limits

• What is a limit?
  • A limit describes the value that a function approaches as the input approaches a certain point.
• Common forms of limits:
  • Some limits can result in indeterminate forms, such as 0/0. Recognizing these is the first step in solving them.

Step 2: Direct Substitution Method

• When to use direct substitution:
  • Use this method if substituting the limit point into the function does not yield an indeterminate form.
• Steps to apply direct substitution:
  1. Identify the function and the point at which you want to find the limit.
  2. Substitute the limit point directly into the function.
  3. Simplify the expression if necessary.
  4. The result will be the limit if it is a defined number.

Example:

• For the function ( f(x) = 2x + 3 ) and finding ( \lim_{x \to 1} f(x) ):
  • Substitute ( x = 1 ):
  • ( f(1) = 2(1) + 3 = 5 ).
  • Thus, ( \lim_{x \to 1} f(x) = 5 ).

Step 3: Factoring Method

• When to use factoring:
  • Use this method when direct substitution results in an indeterminate form like 0/0.
• Steps to apply factoring:
  1. Identify the function and point of interest.
  2. Substitute the limit point to check if it results in 0/0.
  3. Factor the numerator and denominator where possible.
  4. Cancel any common factors.
  5. Use direct substitution on the simplified expression to find the limit.

Example:

• For the function ( f(x) = \frac{x^2 - 1}{x - 1} ) and finding ( \lim_{x \to 1} f(x) ):
  1. Substitute ( x = 1 ): ( f(1) = \frac{1^2 - 1}{1 - 1} = \frac{0}{0} ), which is indeterminate.
  2. Factor the numerator: ( x^2 - 1 = (x - 1)(x + 1) ).
  3. Rewrite the function: ( f(x) = \frac{(x - 1)(x + 1)}{x - 1} ).
  4. Cancel the common factor: ( f(x) = x + 1 ) (for ( x \neq 1 )).
  5. Now substitute ( x = 1 ): ( f(1) = 1 + 1 = 2 ).
  • Thus, ( \lim_{x \to 1} f(x) = 2 ).

Conclusion

In this tutorial, you learned how to find limits in algebra using both direct substitution and factoring methods. The key is to identify when to use each method and how to simplify expressions effectively. As a next step, practice these techniques with different functions to solidify your understanding and improve your problem-solving skills in calculus.