Functions and Limits

Unit 01: Functions and Limits

Notes (Solutions) of Unit 01:Functions and Limits, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. 

Contents & summary

• Introduction
  • Concept of Function
  • Definition (Function-Domain-Range)
  • Notation and Values of a Function
  • Graphs of Algebraic functions
  • Graph of Functions Defined Piece-Wise
• Types of Functions
  • Algebraic Function
  • Trigonometric Functions
  • Inverse Trigonometric Functions
  • Inverse Trigonometric Functions
  • Exponential Function
  • Logarithmic Function
  • Hyperbolic Function
  • Inverse Hyperbolic Function
  • Explicit Function
  • Even Function
  • Odd Function
  • Exercise 1.1
• Composition of Function and Inverse of a Function
  • Composition of Functions Explanation
  • Inverse of a Function
  • Algebraic Method to find the Inverse Function
  • Exercise 1.2
• Limits of a Function and Theorems on Limits
  • Meaning of the phrase “x approaches Zero”
  • Meaning of the Phrase “x approaches a”
  • Concept of Limit of a Function
  • Limit of Function
  • Theorems on Limits of Function
• Limits of Important Functions
  • limx→axn−anx−a=nan−1, where n is an integer and a>0
  • limx→0x+a√−a√x=12a√
  • Limit at Infinity
  • Methods for Evaluating the limits at Infinity
  • limx→0(1+1n)n=e
  • limx→0ax−1x=logea
  • The Sandwitch theorem
  • If θ is measured in radian, then limθ→0sinθθ=1
  • Exercise 1.3
• Continuous and Discontinuous Function
  • One-sided Limits
  • Criterion for Existence of Limit of a Function
  • Continuity of a function at a number
  • Exercise 1.4
• Graphs
• Graph of the Exponential Function f(x)=ax
• Graph of the Exponential Function f(x)=ex
• Graph of Common Logarithmic Function f(x)=logx
• Graph of natural logarithmic Function f(x)=lnx
• Graph of Implicit Function
• Graph of parametric Equations
• Graph of Discontinuous Function
• Graphical Solution of the Equations
• Exercise 1.5

• Exercise 1.1
• Exercise 1.2