Limits and Derivatives — Class 11 MCQs with Answers

Limits and Derivatives — important questions & MCQs with answers (Class 11 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   lim(x→2) (x²) =

   A∞B2C0D4
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   Answer: 4

   Explanation: For a continuous function such as x², the limit at a point equals the value there: substitute x = 2 to get 2² = 4. Writing just x = 2 as the answer confuses the input with the limit.

2. Q2 of 6Easy

   d/dx (xⁿ) =

   AxⁿBn·xⁿ⁻¹CnxDxⁿ⁺¹/(n+1)
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   Answer: n·xⁿ⁻¹

   Explanation: Power rule for derivatives: bring the exponent down as a coefficient and reduce it by one, giving n·xⁿ⁻¹. Leaving xⁿ unchanged or forgetting to drop the exponent is the usual slip.

3. Q3 of 6Medium

   lim (x→2) (x² − 4)/(x − 2) =

   A4B0C2DUndefined
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   Answer: 4

   Explanation: Direct substitution gives 0/0, so factor: (x² − 4)/(x − 2) = (x + 2)(x − 2)/(x − 2) = x + 2, which → 4 as x → 2. Stopping at the 0/0 form and calling the limit undefined is the trap.

4. Q4 of 6Medium

   Assertion (A): The derivative of x² cos x is 2x cos x − x² sin x. Reason (R): For functions u and v, (uv)′ = u′v + uv′.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true and R is the correct explanation of A

   Explanation: A is true: with u = x² and v = cos x, u′ = 2x and v′ = −sin x, so (uv)′ = 2x cos x + x²(−sin x). R is the product rule, which is true and gives A directly.

5. Q5 of 6Hard

   The value of the derivative of (x² + 1)/(x − 1) at x = 2 is:

   A3B1C−1D5
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   Answer: −1

   Explanation: The quotient rule gives (x² − 2x − 1)/(x − 1)², which at x = 2 equals −1.

6. Q6 of 6Hard

   Assertion (A): If f(x) = 2x² + 3x − 5, then f′(0) + 3f′(−1) = 0. Reason (R): The limit of a quotient of two functions is the quotient of their limits, whenever the limit of the denominator is non-zero.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: A is true: f′(x) = 4x + 3, so f′(0) = 3 and f′(−1) = −1, giving 3 + 3(−1) = 0. R is also true, but A comes from the derivative of a polynomial, not from the quotient rule for limits, so R does not explain A.

Limits and Derivatives notes: key concepts (Class 11 Mathematics)

A limit captures the value a function heads toward as its input nears a point, even where the function is undefined. This chapter develops the algebra of limits, the standard trigonometric, exponential and algebraic limits, and the derivative as a limit — the instantaneous rate of change — with rules for fast differentiation.

Intuitive limit

lim(x→a) f(x) = L means f(x) can be made as close to L as we like by taking x close enough to a, whether or not f(a) itself exists.

One-sided limits

The left-hand limit lets x approach a from below and the right-hand limit from above; each describes the function's behaviour on one side only.

Existence of a limit

A limit exists only when the left- and right-hand limits are both finite and equal; a jump between them means the limit fails to exist.

Algebra of limits

The limit of a sum, difference, product or quotient is that same operation on the individual limits, provided each exists (and the denominator ≠ 0).

Limits by substitution

For polynomials, and rationals whose denominator is non-zero at a, the limit is found by directly substituting x = a into the expression.

0/0 indeterminate form

When substitution gives 0/0, factorise, rationalise or apply a standard limit to cancel the vanishing factor before evaluating again.

Standard trigonometric limit

lim(x→0)(sin x)/x = 1 provided x is in radians; from it also follow (tan x)/x → 1 and (1 − cos x)/x → 0 as x → 0.

Standard exponential limits

lim(x→0)(eˣ − 1)/x = 1 and lim(x→0) log(1 + x)/x = 1 are the building blocks for exponential and logarithmic limits.

Standard algebraic limit

lim(x→a)(xⁿ − aⁿ)/(x − a) = n·aⁿ⁻¹ holds for every rational n and is the limit form that underlies the power rule.

Derivative (first principles)

f′(x) = lim(h→0)[f(x+h) − f(x)]/h measures the instantaneous rate of change and the slope of the tangent to the curve.

Rules of differentiation

Differentiation is linear: (cf)′ = cf′ and (f ± g)′ = f′ ± g′; products and quotients need the dedicated product and quotient rules.

Power rule

d/dx(xⁿ) = n·xⁿ⁻¹ for any rational n; combined with linearity it differentiates every polynomial term by term.

Derivatives of trig functions

d/dx(sin x) = cos x and d/dx(cos x) = −sin x, both derived from first principles using the standard trigonometric limit.