Limits Concept

Grade 12 Β· calculus Β· 100 practice problems Β· read aloud

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Limits: The Foundation of Calculus

1. What is a Limit? πŸ€”

A limit describes the value that a function approaches as the input (x) approaches some value. It's the fundamental concept behind calculus, allowing us to define instantaneous rates of change (derivatives) and areas under curves (integrals). We use the notation: limx→a f(x) = L.

2. How to Find a Limit: A Step-by-Step Guide

  1. Step 1: Direct Substitution. Plug the value a into the function. If you get a real number, that's your limit!
  2. Step 2: Indeterminate Form (0/0). If substitution gives 0/0, you need to simplify the function.
  3. Step 3: Simplify. Try factoring, expanding, or rationalizing to cancel the term causing the zero.
  4. Step 4: Substitute Again. After simplifying, substitute the value a again to find the limit.

3. Worked Examples

Example 1: Direct Substitution

Find limx→3 (2x + 1)

Step 1: Substitute x = 3: 2(3) + 1 = 7.

Answer: The limit is 7.

Example 2: Factoring to Resolve 0/0

Find limx→2 (x² - 4)/(x - 2)

Step 1: Substitute: (4 - 4)/(2 - 2) = 0/0 (Indeterminate).

Step 2: Factor the numerator: (x - 2)(x + 2)/(x - 2).

Step 3: Cancel the (x - 2) terms, leaving (x + 2).

Step 4: Substitute x = 2: 2 + 2 = 4.

Answer: The limit is 4.

4. Common Mistakes to Avoid 🚫

Mistake 1: Thinking 0/0 equals 1 or 0. It's undefined and means you need to do more work!

Mistake 2: Cancelling terms before checking for indeterminate form. Always try substitution first.

Mistake 3: Confusing the value a function approaches with the value it actually is at a point (if it's even defined there).

5. Tips & Tricks

  • Memory Aid: "Hole-y Cow!" A limit helps you find the y-value of a "hole" in a graph.
  • Shortcut: For rational functions, if direct substitution fails, factoring is almost always the first tool to try.
  • Strategy: If you see a radical, consider multiplying by the conjugate.

6. How to Practice

Master limits with these steps:

  1. Start with simple polynomials to master direct substitution.
  2. Move to rational functions and practice factoring.
  3. Tackle problems with radicals that require rationalizing.
  4. Challenge yourself with piecewise functions and graphs to interpret limits visually.

Practice problems

6 of the 100, worked through step by step β€” try them before opening the answer.

1 From graph: as x→7, f(x)→?

Hint: Observe the y-values as x gets closer to 7 from both the left and right sides on the graph.

Show the answer

Answer: 5

  1. Look at the graph and find the point where x approaches 7.
  2. Check the y-values as x approaches 7 from the left (x = 6.9, 6.99, 6.999). The y-values get closer to 5.
  3. Check the y-values as x approaches 7 from the right (x = 7.1, 7.01, 7.001). The y-values also get closer to 5.
  4. Since the y-values approach 5 from both sides, the limit is 5.

The answer is 5.

2 From graph: as x→8, f(x)→?

Hint: Examine how the y-values behave as x gets closer to the given value from both sides

Show the answer

Answer: 5

  1. Look at the graph and identify the point where x approaches 8
  2. Observe the y-values as x approaches 8 from the left side (x = 7.9, 7.99, 7.999)
  3. Observe the y-values as x approaches 8 from the right side (x = 8.1, 8.01, 8.001)
  4. Notice that from both sides, the y-values approach 5
  5. Since the y-values approach the same number from both sides, the limit exists and equals 5

The answer is 5.

3 From graph: as x→1, f(x)→?

Hint: Observe the y-values that f(x) approaches as x gets closer to 1 from both the left and right sides on the graph.

Show the answer

Answer: 6

  1. Look at the graph and find the point where x is approaching 1.
  2. Check the y-values as x approaches 1 from the left side (x values like 0.9, 0.99). The y-values get closer to 6.
  3. Check the y-values as x approaches 1 from the right side (x values like 1.1, 1.01). The y-values also get closer to 6.
  4. Since both the left-hand and right-hand limits approach 6, the limit as x approaches 1 is 6.

The answer is 6.

4 From graph: as x→5, f(x)→?

Hint: Observe the y-values on the graph as x gets closer to the given value from both sides

Show the answer

Answer: 15

  1. Look at the graph and find where x approaches 5 from the left side
  2. As x approaches 5 from the left, the y-values approach 15
  3. Look at the graph and find where x approaches 5 from the right side
  4. As x approaches 5 from the right, the y-values also approach 15
  5. Since both left-hand and right-hand limits approach the same value, the limit exists
  6. Therefore, lim(x→5) f(x) = 15

5 lim(x→0) (sin(3x))/(2x) = ?

Hint: Consider the standard limit result for sin(ΞΈ)/ΞΈ as ΞΈ approaches 0, and how to manipulate the given expression to match that form.

Show the answer

Answer: 1.5

  1. Recall the standard limit: lim(ΞΈβ†’0) sin(ΞΈ)/ΞΈ = 1
  2. We have lim(x→0) sin(3x)/(2x)
  3. Multiply numerator and denominator by 3/3: (3/3) Γ— sin(3x)/(2x) = (3 sin(3x))/(6x)
  4. Rewrite as (3/2) Γ— (sin(3x))/(3x)
  5. As x→0, 3x→0, so (sin(3x))/(3x) → 1
  6. Therefore, the limit equals (3/2) Γ— 1 = 3/2 = 1.5

The answer is 1.5.

6 lim(x→0) (e^(2x) - 1)/x = ?

Hint: Consider the derivative definition of the exponential function at x=0

Show the answer

Answer: 2

  1. Recognize that this limit matches the derivative definition f'(0) = lim(x→0) [f(x) - f(0)]/x
  2. Let f(x) = e^(2x), so f(0) = e^0 = 1
  3. The derivative f'(x) = 2e^(2x)
  4. Evaluate at x=0: f'(0) = 2e^0 = 2
  5. Therefore, lim(x→0) (e^(2x) - 1)/x = f'(0) = 2

The answer is 2.

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