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An intuitive introduction to the concept of limits in calculus
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A limit describes what happens to a function as the input gets closer and closer to a certain value. Think of it as asking: "Where is this function heading?"
Key Idea: Limits are about the journey, not the destination. We care about where a function is going, not necessarily where it is.
We write:
Using your intuition, estimate by checking values near x = 2.
| Section | Format | Questions | Time | Weight | Calculator |
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| Multiple Choice (No Calculator) | MCQ | 30 | 60 min | 33.3% | 🚫 |
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| Free Response (Calculator) | FRQ | 2 | 30 min | 16.7% | ✅ |
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This reads: "The limit of f(x) as x approaches a equals L"
Consider the function:
What is ?
As x gets close to 3:
The values are getting closer to 7!
Therefore:
Limits are the foundation of calculus because they help us:
The limit of f(x) as x approaches a might be different from f(a), or f(a) might not even exist!
Let's check values approaching 2 from both sides:
From the left (x < 2):
From the right (x > 2):
Both sides approach 5, so:
What does lim(x→2) f(x) = 5 mean in words?
Step 1: Break down the notation: • lim means "limit" • x→2 means "as x approaches 2" • f(x) is the function • = 5 is the value the function approaches
Step 2: Interpret: As x gets closer and closer to 2 (from both sides), the function values f(x) get closer and closer to 5
Step 3: Important note: This does NOT require f(2) = 5 The limit describes behavior NEAR x = 2, not AT x = 2
Answer: As x approaches 2, f(x) approaches 5
If f(x) = (x² - 4)/(x - 2), describe what happens to f(x) as x approaches 2.
Step 1: Note that f(2) is undefined: f(2) = (4 - 4)/(2 - 2) = 0/0 (indeterminate)
Step 2: Factor the numerator: f(x) = (x² - 4)/(x - 2) = [(x - 2)(x + 2)]/(x - 2)
Step 3: Simplify for x ≠ 2: f(x) = x + 2
Step 4: Evaluate the limit: As x→2, f(x) = x + 2 → 2 + 2 = 4
Step 5: Create a table to verify: x: 1.9 1.99 1.999 →2← 2.001 2.01 2.1 f(x): 3.9 3.99 3.999 4 4.001 4.01 4.1
Answer: As x approaches 2, f(x) approaches 4
Consider f(x) = |x|/x. What is lim(x→0) f(x)?
Step 1: Analyze from the left (x < 0): When x < 0, |x| = -x f(x) = -x/x = -1
Step 2: Analyze from the right (x > 0): When x > 0, |x| = x f(x) = x/x = 1
Step 3: Check if left and right limits agree: lim(x→0⁻) f(x) = -1 lim(x→0⁺) f(x) = 1
Step 4: Since left ≠ right: The limit does NOT exist
Step 5: Conclusion: lim(x→0) f(x) does not exist (DNE)
Answer: The limit does not exist
Explain why a function can have a limit at x = a even if f(a) is undefined.
Step 1: Understand what a limit measures: A limit describes the behavior of f(x) as x APPROACHES a, not the value AT x = a
Step 2: Example: f(x) = (x² - 1)/(x - 1) • f(1) is undefined (0/0) • But lim(x→1) f(x) = 2
Step 3: Why this works: Factor: f(x) = [(x - 1)(x + 1)]/(x - 1) = x + 1 for x ≠ 1 As x→1, f(x)→2, even though f(1) doesn't exist
Step 4: The key concept: Limits care about NEARBY values, not the value AT the point We can get arbitrarily close to x = a without ever reaching it
Step 5: Practical meaning: This is why limits are useful for defining derivatives (which often involve 0/0 forms)
Answer: Limits describe nearby behavior, not the value at the point itself