Limits and Derivatives — Practice

Question 1

Evaluate lim(x->2) (x^2-4)/(x-2).

Accepted Answer

Direct substitution gives 0/0. Factor: x^2-4=(x-2)(x+2), so the expression equals x+2 for x!=2. Answer: The limit is 4.

Question 2

Evaluate lim(x->0) sin(5x)/sin(3x).

Accepted Answer

Write as [sin(5x)/5x . 5] / [sin(3x)/3x . 3] = (5/3).[sin(5x)/5x]/[sin(3x)/3x]. As x->0, both bracketed ratios tend to 1. Answer: The limit is 5/3.

Question 3

For f(x) = 3x+1 if x<2, and f(x)=x^2+3 if x>=2, determine whether lim(x->2) f(x) exists.

Accepted Answer

Left-hand limit: 3(2)+1=7. Right-hand limit (using x>=2 branch at x=2): 2^2+3=7. Answer: Since both one-sided limits equal 7, the limit exists and equals 7.

Question 4

Find the derivative of f(x) = x^3 from first principles.

Accepted Answer

f'(x) = lim(h->0) [(x+h)^3-x^3]/h. Expanding: (x+h)^3=x^3+3x^2h+3xh^2+h^3, so the difference is 3x^2h+3xh^2+h^3. Dividing by h: 3x^2+3xh+h^2. Answer: Taking the limit as h->0 gives f'(x) = 3x^2.

Question 5

Find the derivative of f(x) = sqrt(x) from first principles (for x>0).

Accepted Answer

f'(x) = lim(h->0) [sqrt(x+h)-sqrt(x)]/h. Rationalise by multiplying by [sqrt(x+h)+sqrt(x)]/[sqrt(x+h)+sqrt(x)]: the numerator becomes (x+h)-x=h, so the expression simplifies to 1/[sqrt(x+h)+sqrt(x)]. Answer: Taking the limit as h->0 gives f'(x) = 1/(2.sqrt(x)).

Question 6

Differentiate y = x^2.sin(x) using the product rule, then evaluate dy/dx at x=pi/2.

Accepted Answer

Using u=x^2, v=sin(x): dy/dx = 2x.sin(x) + x^2.cos(x). At x=pi/2: 2(pi/2)(1) + (pi/2)^2(0) = pi. Answer: dy/dx = 2x sin(x) + x^2 cos(x), which equals pi at x=pi/2.

Question 7

Differentiate y = (x^2+1)/(x-1) using the quotient rule.

Accepted Answer

Using u=x^2+1, u'=2x, v=x-1, v'=1: dy/dx = [2x(x-1)-(x^2+1)(1)]/(x-1)^2 = [2x^2-2x-x^2-1]/(x-1)^2 = (x^2-2x-1)/(x-1)^2. Answer: dy/dx = (x^2-2x-1)/(x-1)^2.

Question 8

Differentiate y = sec(x) using the quotient rule on 1/cos(x).

Accepted Answer

Using u=1, u'=0, v=cos(x), v'=-sin(x): dy/dx = [0.cos(x) - 1.(-sin(x))]/cos^2(x) = sin(x)/cos^2(x) = (1/cos x).(sin x/cos x). Answer: dy/dx = sec(x).tan(x).

Question 9

Differentiate f(x) = (3x+2)^5 using the first-principles substitution technique (without invoking a general chain rule).

Accepted Answer

Using d/dx[(ax+b)^n] = a.n.(ax+b)^(n-1) with a=3, b=2, n=5: derivative = 3.5.(3x+2)^4. Answer: f'(x) = 15(3x+2)^4.

Question 10

Evaluate lim(x->0) (e^(3x)-1)/x, using the standard exponential limit.

Accepted Answer

Write as 3.[e^(3x)-1]/(3x). As x->0, 3x->0 too, and the bracketed ratio tends to 1 by the standard limit lim(t->0)(e^t-1)/t=1. Answer: The limit is 3.

Practice — Limits and Derivatives

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