EN112_Slide_Show_Limits

EN112 Engineering Mathematics I - Functions and Limits

Course Overview

• Course Title: EN112 Engineering Mathematics I

• Department: Mathematics and Computer Science

• Institution: Papua New Guinea University of Technology

• Instructor: Dr. Dunstan

• Location: Room MCS213, Mathematics and Computer Science Building

• Semester: Semester 1, 2025

Introduction to Limits

• Definition of Limits: The fundamental concept in calculus that is essential for all calculus operations in physics and engineering.

  • Informal Explanation:

    • If f(x) can be made close to a value L by selecting x values close to a (not equal to a), we express this as:

      [ \lim_{x \to a} f(x) = L ]

    • This expression is read as "The limit of f(x) as x approaches a is L."

• Illustration: The limit as average velocity approaches instantaneous velocity.

Specific Limits and Asymptotes

• Example of Limits:

  • [ \lim_{x \to 0} \frac{\sin x}{x} = 1 ]

  • This limit shows that simply substituting 0 for x may lead to the wrong conclusion.

• Vertical Asymptotes:

  • Occur when ( f(x) \to +\infty ) or ( f(x) \to -\infty ) as ( x \to a ) from either direction.

• Horizontal Asymptotes:

  • Defined by the conditions:

    • ( \lim_{x \to \infty} f(x) = L )

    • ( \lim_{x \to -\infty} f(x) = L )

Continuity of Functions

• A function f is continuous at x = c if:

  1. f(c) is defined.

  2. ( \lim_{x \to c} f(x) ) exists.

  3. ( \lim_{x \to c} f(x) = f(c) )

Indeterminate Forms of Limits

• Indeterminate Form Example:

  • For ( \lim_{x \to 0} \frac{\sin x}{x} ): initially appears indeterminate.

  • Another Example: ( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} ):

    • Factorization leads to:[ \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = 4 ]

• Methods of Solving Indeterminate Forms:

  • Graphical/geometric visualization.

  • Series expansion.

  • Differentiation using L'Hopital's rule.

L’Hopital’s Rule

• L'Hopital's Rule: Used for evaluating limits of the form ( \frac{0}{0} ).

• Derivation:

  • If ( f(x) = f(a+h) ) and ( g(x) = g(a+h) ):

    • We derive:[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{h \to 0} \frac{f(a+h)}{g(a+h)} = \lim_{h \to 0} \frac{tanPAK}{tanQAK} = \frac{f'(a)}{g'(a)} ]

  • This leads to the conclusion:[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} ]

Application of L’Hopital’s Rule

• Conditions for Application:

  • f and g are differentiable on an interval except at x = a.

  • Both limits ( \lim_{x \to a} f(x) = 0 ) and ( \lim_{x \to a} g(x) = 0 ).

• Note: Check that the limit is truly indeterminate before applying L'Hopital's Rule.