Latihan Soal dan Pembahasan Kalkulus 1 : Limit dan Kekontinuan

Introduction

This tutorial focuses on the foundational concepts of limits and continuity in calculus, as presented in the video "Latihan Soal dan Pembahasan Kalkulus 1: Limit dan Kekontinuan" by Rina Mardiati. Understanding these concepts is crucial for mastering calculus and applying it to various mathematical problems. This guide simplifies the key points and problems discussed in the video, helping you grasp the essentials of limits and continuity.

Step 1: Understanding Limits

• Definition of a Limit: A limit is a value that a function approaches as the input approaches a certain point.
• Notation: Limits are typically denoted as:
  • ( \lim_{x \to c} f(x) = L )
  • This means as ( x ) approaches ( c ), ( f(x) ) approaches ( L ).

Practical Tips

• To calculate limits, substitute the value of ( x ) directly into the function when possible.
• If direct substitution results in an indeterminate form (like ( 0/0 )), apply algebraic manipulations or limit laws.

Step 2: Evaluating Limits with Examples

• Example 1: Find ( \lim_{x \to 2} (3x + 1) )

  • Substitute ( x = 2 ):
    • ( 3(2) + 1 = 7 )
  • Thus, ( \lim_{x \to 2} (3x + 1) = 7 ).

• Example 2: Find ( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} )

  • Direct substitution gives ( 0/0 ).
  • Factor the numerator: ( \frac{(x - 3)(x + 3)}{x - 3} ).
  • Simplify to ( x + 3 ) for ( x \neq 3 ).
  • Now substitute ( x = 3 ):
    • ( 3 + 3 = 6 )
  • Thus, ( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} = 6 ).

Step 3: Understanding Continuity

• Definition of Continuity: A function is continuous at a point ( c ) if:
  • ( f(c) ) is defined.
  • ( \lim_{x \to c} f(x) ) exists.
  • ( \lim_{x \to c} f(x) = f(c) ).

Common Pitfalls

• Ensure that the function is defined at the point of interest.
• Check that the limit from both sides is equal to the function value at that point.

Step 4: Evaluating Continuity with Examples

• Example 1: Check if ( f(x) = \frac{x^2 - 1}{x - 1} ) is continuous at ( x = 1 ).

  • Simplify ( f(x) ): ( f(x) = x + 1 ) for ( x \neq 1 ).
  • ( f(1) ) is not defined; thus, it is not continuous at ( x = 1 ).

• Example 2: Check if ( g(x) = 2x + 1 ) is continuous at ( x = 1 ).

  • ( g(1) = 3 ), and ( \lim_{x \to 1} g(x) = 3 ).
  • Since both conditions are met, ( g(x) ) is continuous at ( x = 1 ).

Conclusion

Understanding limits and continuity is essential for progressing in calculus. This tutorial covered the definitions, evaluation techniques, and examples that illustrate these concepts. To advance your learning:

• Practice more limit problems using various functions.
• Explore different functions to determine their continuity.
• Review the definitions and properties regularly to reinforce your understanding.

Continue your studies and apply these concepts in more complex calculus problems for better mastery!