Introduction to Limits

Key Concepts

• Limit: A value that a function approaches as the input approaches some value.
• Direct Substitution: Plugging in the value of x directly into the function to find the limit.
• Indeterminate Forms: Forms like 0/0 that need further analysis to evaluate.

Evaluating Limits

Example 1

Limit as x approaches 2 of (x^2 - 4)/(x - 2):

• Direct substitution: Results in 0/0 which is undefined.
• Try values close to 2 (e.g., 2.1): The limit approaches 4.
• Factorization method: (x^2 - 4) = (x + 2)(x - 2).
  • Cancel out (x - 2).
  • Evaluate remaining expression x + 2: Result is 4.

Example 2

Limit as x approaches 5 of x^2 + 2x - 4:

• Direct substitution works: Evaluate as 5^2 + 2(5) - 4 = 31.

Example 3

Limit as x approaches 3 of (x^3 - 27)/(x - 3):

• Direct substitution gives 0/0.
• Factorize x^3 - 27: Use the difference of cubes formula.
  • (x^3 - 27) = (x - 3)(x^2 + 3x + 9).
  • Cancel out (x - 3) and evaluate remaining expression: Result is 27.

Example 4

Limit as x approaches 3 of 1/x - 1/3 / (x - 3):

• Multiply by the common denominator 3x on top and bottom.
• Simplify and cancel terms.
• Direct substitution: Result is -1/9.

Example 5

Limit as x approaches 9 of (sqrt(x) - 3) / (x - 9):

• Multiply by the conjugate sqrt(x) + 3.
• Simplify and cancel terms.
• Direct substitution: Result is 1/6.

Example 6

Limit as x approaches 4 of 1/sqrt(x) - 1/2 / (x - 4):

• Multiply by 2sqrt(x) and by the conjugate.
• Factor out -1 and simplify.
• Direct substitution: Result is -1/16.

Evaluating Limits Graphically

Key Points

• Identify x value and approach it from the left/right on the graph.
• The limit is the y value the function approaches.
• If left and right limits are not the same, the limit does not exist.

Example 1

Limit as x approaches -3:

• From the left: y approaches 1.
• From the right: y approaches -3.
• Conclusion: Limit does not exist as the values do not match.

Illustration of One-Sided Limits and Function Values

• One-sided limits (values approaching from left or right).
• Function values at specific points (closed circles).

Example 2

Limit as x approaches -2:

• From the left and right: y value is -2.
• Middle limit exists and equals -2.
• Function value at -2: y is 2 (based on closed circle).

Example 3

Limit as x approaches 1:

• From the left: y value is 1.
• From the right: y value is 3.
• Middle limit does not exist as values do not match.
• Function value at 1: y is 2 (closed circle).

Example 4

Limit as x approaches 3:

• As x approaches 3: Exhibit a vertical asymptote.
  • From the left: y goes to negative infinity.
  • From the right: y goes to positive infinity.
• Middle limit does not exist. Function value is undefined.

Types of Discontinuities

• Jump Discontinuity: Non-removable; graph “jumps” at a specific point.
• Hole: Removable discontinuity; small gap in the graph.
• Infinite Discontinuity: Caused by vertical asymptotes; non-removable.