b. If f(x) = 1/x then the derivative of with respect to x is f(x) with respect to x is 1/(x2)
C. The limit of (ex - 1)/x as x approaches 0 is 1.
d. The limit definition of a derivative for a function f(x) is
Which of the following statement is correct?
Last updated Jun 24, 2026
Correct Answer:
Option A —
b and c
Verification of each statement:
a. The limit of [blank/missing expression] as x approaches 1 is 0.
Incorrect / cannot be evaluated as correct.
The statement is incomplete (the function or expression whose limit is being taken is missing). Without the specific function, we cannot confirm if the limit is indeed 0 as x → 1. Many common limits as x → 1 are not 0 (e.g., lim (x-1)/(x-1) is 1, lim 1/(x-1) does not exist, etc.). Since it's malformed, statement a is not correct.
b. If f(x) = 1/x then the derivative of f(x) with respect to x is 1/(x²).
Incorrect as written (sign error).
The actual derivative is −1/x².
Using power rule: derivative of x⁻¹ is −1 ⋅ x⁻² = −1/x².
Using first principles (limit definition) also yields −1/x².
The statement misses the negative sign, so b is false.
c. The limit of (eˣ - 1)/x as x approaches 0 is 1.
Correct.
This is a standard limit:
$$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$$
It can be proven using L'Hôpital's rule (0/0 form → derivative of numerator eˣ, denominator 1 → e⁰/1 = 1), Taylor series, or the definition of the derivative of eˣ at x=0.
d. The limit definition of a derivative for a function f(x) is [blank/missing].
