A student computes f'(a) for f(x) = x² using the limit definition. After forming the difference quotient [(a+h)² − a²]/h, they immediately substitute h = 0 and get 0/0. What went wrong?
AThey should have used h→∞ instead of h→0
BThey needed to simplify the expression algebraically before evaluating the limit
CThe derivative of x² doesn't exist, so 0/0 is the correct result
DThey should have computed (f(a) − f(a−h))/h instead
Substituting h = 0 directly yields the indeterminate form 0/0, which provides no information. The algebra must be done first — expand (a+h)², cancel the a² terms, factor out h from the numerator, and cancel it with the denominator. Only then can you evaluate at h = 0 to get the derivative. The entire point of the limit definition is that you simplify until the indeterminate form is resolved, then take the limit.
Question 2 Multiple Choice
A function f is continuous at x = a but not differentiable there. Which example best illustrates this?
Af(x) = x² at x = 0
Bf(x) = |x| at x = 0
Cf(x) = 1/x at x = 0
Df(x) = sin(x) at x = 0
f(x) = |x| is continuous at x = 0 (no gap or jump), but the left-hand difference quotient approaches −1 while the right-hand difference quotient approaches +1. Since the one-sided limits disagree, the limit defining f'(0) does not exist — the derivative fails. This is the canonical example showing that continuity does not imply differentiability. Note: f(x) = 1/x is not even continuous at 0, so it doesn't illustrate the right distinction.
Question 3 True / False
If the limit f'(a) = lim_{h→0} [f(a+h) − f(a)]/h exists, then f must be continuous at x = a.
TTrue
FFalse
Answer: True
Differentiability implies continuity — this is a theorem, not just a rule of thumb. If the difference quotient has a finite limit, the numerator f(a+h) − f(a) must approach 0 as h → 0 (since the denominator h also approaches 0 and a finite ratio requires the numerator to vanish). That means f(a+h) → f(a), which is exactly the definition of continuity at a. The converse fails: continuous functions need not be differentiable (e.g., |x| at 0).
Question 4 True / False
The difference quotient [f(a+h) − f(a)]/h evaluates to 0/0 when h = 0, which means the derivative at a equals 0.
TTrue
FFalse
Answer: False
0/0 is an indeterminate form — it does not equal 0 or any other specific number. It signals that direct substitution cannot determine the limit; the expression must be algebraically simplified first. For example, for f(x) = x², the difference quotient simplifies to 2a + h, and only then substituting h = 0 gives 2a — which is typically nonzero. The derivative is determined by the limit after simplification, not by the indeterminate form before it.
Question 5 Short Answer
Why is it necessary to simplify the difference quotient algebraically before taking the limit, rather than substituting h = 0 directly?
Think about your answer, then reveal below.
Model answer: Direct substitution gives 0/0 — an indeterminate form with no value. The algebraic work (expanding and simplifying the numerator) allows h to cancel from numerator and denominator, converting the expression into a form that can be evaluated at h = 0. The limit process works precisely because the cancellation removes the problematic h in the denominator before it reaches zero.
This is the computational heart of the limit definition. The difference quotient is designed to produce an indeterminate form — because you're trying to find the slope at a single point, which formally requires dividing zero by zero. The algebra resolves the indeterminacy by revealing what the expression approaches. Understanding this prevents the most common mechanical error in limit computations: premature substitution.