#14: LIMITS AT INFINITY AGAIN - CURVE SKETCHING

LIMITS AT INFINITY

Methods for evaluating f (x) :

NUMERIC METHODS:

• Tables--Make a table of x and f (x) for larger and larger values of x and observe if the f (x) values are approaching a single finite value. This will be the value of the limit.
• Graphs--Graph the function for very large values of x. Look for horizontal asymptotes. The y value of the horizontal asymptote is the value of the limit.
• The above two methods can also be used for f (x), except the x values are large in magnitude but negative in sign.
• Note: Just as with finite limits, numerical methods do not prove the value of a limit; but they allow you to approximate the limit. Also, for very large values of x you may exceed the capacity of your calculator or computer to do the necessary computations.

1. Use a table to evaluate

a)        b)

2. TRY IT #1: Use a table to evaluate

a)        b)

3. Use a table to evaluate

a)        b)

4. TRY IT #2: Use a table to evaluate

a)        b)

5. Use a table to evaluate

a) ( cos x ) / x ²        b) ( cos x ) / x ²

6. TRY IT #3: Use a table to evaluate

a) ( tan ² x ) / x ²        b) ( tan ² x ) / x ²

7. Use a graph to evaluate

a)        b)

8. TRY IT #4: Use a graph to evaluate

a)        b)

9. Use a graph to evaluate

a)        b)

10. TRY IT #5: Use a graph to evaluate

a)        b)

11. Use a graph to evaluate

a) ( cos x ) / x ²        b) ( cos x ) / x ²

12. TRY IT #6: Use a graph to evaluate

a) x e ^{- x}        b) x e ^{- x}

ALGEBRAIC METHODS:

Some helpful limits to memorize are

• 1 / x ^r = 0 if r > 0
• 1 / x ^r = 0 if r > 0 and x ^r is defined
• e ^{- x} = 0
• e ^x = 0

13. Evaluate the following limits algebraically

1. 

1. What happens when you use the quotient rule for limits?
2. Divide numerator and denominator by e ^x and simplify.
3. Now what happens when you use the quotient rule for limits?
2. 

14. TRY IT #7: Evaluate the following limits algebraically

1. 
2. 

1. What happens when you use the quotient rule for limits?
2. Multiply numerator and denominator by e ^x and simplify.
3. Now what happens when you use the quotient rule for limits?

15. Evaluate the following limits algebraically

1. 

1. What happens when you use the quotient rule for limits?
2. Divide numerator and denominator by x and simplify.
3. Now what happens when you use the quotient rule for limits?
2. 

1. What happens when you use the quotient rule for limits?
2. Divide numerator and denominator by x and simplify.
3. Now what happens when you use the quotient rule for limits?

16. TRY IT #8: Evaluate the following limits algebraically

1. 

1. What happens when you use the quotient rule for limits?
2. Divide numerator and denominator by x ² and simplify.
3. Now what happens when you use the quotient rule for limits?
2. 

1. What happens when you use the quotient rule for limits?
2. Divide numerator and denominator by x ² and simplify.
3. Now what happens when you use the quotient rule for limits?

SQUEEZE THEOREM--Find two functions h (x) and g (x) such that

• h (x) <  f (x) < g (x) and
• h and g have the same limit as x increases without bound (or decreases without bound.)

Then this common limit will be the limit of f as x increases (or decreases) without bound.

17. Evaluate the limit algebraically

1. x e ^{- x}

1. What happens when you use the product rule for limits?
2. Use the fact that 1 + x + 0.5 x ² < e ^x to bound the x e ^{- x} between two functions
3. Evaluate the limit.
2. x e ^{- x}

1. What happens when you use the product rule for limits?
2. Evaluate the limit.

18. TRY IT #9: Evaluate the limit algebraically

1. ( cos x ) / x ²

1. What happens when you use the quotient rule for limits?
2. Use the fact that -1 < cos x < 1 to bound the function between two functions
3. Evaluate the limit.
2. ( cos x ) / x ²

1. What happens when you use the quotient rule for limits?
2. Use the fact that -1 < cos x < 1 to bound the function between two functions
3. Evaluate the limit.

CURVE SKETCHING

The following is a list of the steps in sketching the graph of a function by hand.

• Determine the domain of the function and its x and y intercepts
• Differentiate the function and do a sign graph for the derivative.
• Determine the critical points and where the function is increasing and decreasing.
• Differentiate again and do a sign graph for the second derivative.
• Determine the points of inflection and where the function is concave up and concave down.
• Determine any vertical or horizontal asymptotes

19. Sketch by hand the graph of

1. Determine the domain and intercepts.
2. Find the derivative

1. Find the critical points
2. Make a sign graph for the derivative
3. Determine relative extrema
3. Find the second derivative.

1. Make a sign graph for the second derivative.
2. Find the inflection points
4. Find the

1. vertical asymptotes
2. horizontal asymptotes
5. graph the function.

20. TRY IT #10: Sketch by hand the graph of x e ^{- x}

1. Determine the domain and intercepts.
2. Find the derivative

1. Find the critical points
2. Make a sign graph for the derivative
3. Determine relative extrema
3. Find the second derivative.

1. Make a sign graph for the second derivative.
2. Find the inflection points
4. Find the

1. vertical asymptotes
2. horizontal asymptotes
5. graph the function.

21. Sketch by hand the graph of

1. Determine the domain and intercepts.
2. Find the derivative

1. Find the critical points
2. Make a sign graph for the derivative
3. Determine relative extrema
3. Find the second derivative.

1. Make a sign graph for the second derivative.
2. Find the inflection points
4. Find the

1. vertical asymptotes
2. horizontal asymptotes
5. graph the function.

22. TRY IT #11: Sketch by hand the graph of

1. Determine the domain and intercepts.
2. Find the derivative

1. Find the critical points
2. Make a sign graph for the derivative
3. Determine relative extrema
3. Find the second derivative.

1. Make a sign graph for the second derivative.
2. Find the inflection points
4. Find the

1. vertical asymptotes
2. horizontal asymptotes
5. graph the function.