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MCV4U: Calculus & Vectors

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  1. Course Outline
    Course Outline
  2. Unit 1: Rates of Change
    Lesson 1: Reviewing prerequisite skills
    1 Quiz
  3. Lesson 2: Determining rates of change
    1 Quiz
  4. Lesson 3: Determining limits
    1 Quiz
  5. Lesson 4: Using first principles to determine the equation of a tangent
    1 Quiz
  6. Unit 2 : Derivatives
    Lesson 5: Finding derivatives (part A)
    1 Quiz
  7. Lesson 6: Finding derivatives (part B)
    1 Quiz
  8. Lesson 7: Solving related rates problems
    1 Quiz
  9. Lesson 8: Investigating velocity, acceleration and second derivatives
    1 Quiz
  10. Unit 3 : Curve Sketching
    Lesson 9: Exploring the first derivative
    1 Quiz
  11. Lesson 10: Exploring the second derivative
    1 Quiz
  12. Lesson 11: Sketching curves: part A
    1 Quiz
  13. Lesson 12: Sketching curves: part B
    1 Quiz
  14. Unit 4 : Extensions
    Lesson 13: Solving optimization problems
    1 Quiz
  15. Lesson 14: Working with sinusoidal functions
    1 Quiz
  16. Lesson 15: Working with exponential and logarithmic functions
    1 Quiz
  17. Unit 5 : Vectors
    Lesson 16: Using geometric vectors
    1 Quiz
  18. Lesson 17: Investigating Cartesian vectors
    1 Quiz
  19. Lesson 18: Exploring vectors in 3-space
    1 Quiz
  20. Lesson 19: Creating equations of vectors
    1 Quiz
  21. Lesson 20: Investigating lines and planes
    1 Quiz
Lesson 22 of 21
In Progress

Lesson 2: Understanding Limits

By definition a limit is: A function f  has a limit L as x approaches a, written  provided that the values of f(x) get closer and closer to L as x gets closer and closer to a, on both sides of a, but x ‚Ȇ a

Another way to state this definition is to say that a limit is the intended height (where height f is a function in term of time x) of a function

Determining the height of a function 

Example 1:

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This function changes height all throughout its domain.¬† That different ‚Äúx‚ÄĚ values have different heights.¬†

What is the height of the above function when x = 2? 

Solution:

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The function reaches a height of 4 when x = 2.  A limit statement can be made.  which simply means that the function as x approaches 2 is getting closer to the height of 4. 

Functions that don‚Äôt reach their intended heights 

Functions that don‚Äôt reach their intended heights 

Example 2: Can a function have a limit if it can’t reach its intended height? 

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This function appears to be a regular function; however, there is a hole in the middle of the line representing this function. 

When x =2 the function is undefined. 

When we substitute x = 2 into the function the following happens. 

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When Does a Limit Exist? 

A limit exists if you travel along a function from the left side and from the right side towards some specific value of x. In other words, as long as the y-values from the left and right are the same then the limit exists. 

Example 3: What is the Limit of the function shown below? 

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Solution:

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For a limit to exist the left hand limit must equal the right hand limit. 

So, in this example the limit does not exist.  

This is also called a discontinued function.  If the limit did exist it would be called a continuous function. 

Discontinued Function:

Continuous Function:

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Evaluating Limits (substitution)

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Example 4:

Solution:

Substitute 3 in for n.

Therefore,

Note: If you substitute and there is an answer, the limit is the answer. 

Example 5:

Solution:

Factor the numerator first then since a substitution yielded an undefined answer, cancel out common factors to the numerator and denominator and substitute. 

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Therefore,

Example 6: Find

Solution: Conjugates are used to evaluate this question because we must rationalize the numerator (see Lesson 1).   

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Therefore,