Limits and Derivatives with MCQ type Questions

What is Limit ?

 A  limit represents the value that a function approaches output for the input (usually denoted as x) approaches a specific value (usually denoted as a). Mathematically, the limit of a function \(f(x)\) for x  approaches a is denoted as \(\underset{x \to a}{\lim}\) \(f(x)= f(a) \).

What is Derivative ?

The derivative of a function measures how the function changes as its input(Independent variable) changes. It represents the slope of the tangent line to the graph of the function at a particular point. <br>The derivative of f(x) with respect to x is denoted as \(\frac{df}{dx} \) or \(f'(x) \).

Derivative Rules

1.The power rule-

\(\frac{d}{dx}[x^n]=nx^{n-1}\).

2.The Sum and Difference rule-
\(\frac{d}{dx}[f(x) \pm g(x)]= f'(x) \pm g'(x) \).
3.The product rule-

\(\frac{d}{dx}[f(x) . g(x)]\)=\( f'(x).g(x) +f(x).g'(x) \).

4.The quotient rule-

\(\frac{d}{dx}[ \frac{f(x)}{g(x)}]= \frac{f'(x).g(x)-f(x).g'(x)}{(g(x))^2}  \).

5.Chain Rule- The chain rule is used to find the derivative of a composite function. If \(y=f(g(x))\), then
\(\frac{dy}{dx}\) =\(\frac{dy}{ dg} . \frac{dg}{dx} \).

20 MCQ and Answers on Limit and Derivative

  1. What is the limit of \( \lim_{x \to 3} (2x – 1) \)?

    (a) 4  (b) 5   (c) 6   (d) 7

  2. What is the limit of \(\lim_{x \to \pi} (cos(x))\)?

    (a) -2  (b) 0   (c) -1   (d) \(\infty \)

  3. What is the derivative of \(f(x) = 3x^2 – 5x + 2\)?

    (a) \(6x – 5\)  (b) \(3x^3 – 5x^2 + 2x\)   (c)\(6x^2 – 5x\)   (d) \(3x – 5\)

  4. What is the limit of \( \lim_{x \to 0} \frac{\sin(x)}{x} \)?

    (a) 0  (b) 1   (c) \(\infty\)   (d) \(-1\)

  5. What is the limit of \(\lim_{x \to 0} \frac{1 – \cos(x)}{x^2}\)?

    (a) \(\frac{1}{2}\)  (b) 1   (c) 2   (d) 0

  6. What is the derivative of \(f(x) = e^x\)?

    (a) \(e^x\)  (b) \(x \cdot e^x\)   (c) \(1/x\)  (d) \(e^{x-1}\)

  7. What is the limit of \( \lim_{x \to 2} \frac{x^2 – 4}{x – 2} \)?

    (a) 2  (b) 4   (c) 0   (d) 3

  8. What is the derivative of \(f(x) = \ln(x)\)?

    (a)\(\ln(x)\)  (b) \(1/x\)   (c) \(\frac{1}{2x}\)  (d)
    x

  9. What is the limit of
    \( \lim_{x \to \infty} \frac{2x^2 + 3x – 5}{x^2 – 2x + 1} \)?

    (a) 2  (b) 3   (c) 1   (d) 0

  10. What is the derivative of \(f(x) = \cos(x)\)?

    (a) \(-\sin(x)\)  (b) \(\cos(x)\)   (c)\(\frac{1}{x}\)  (d) \(\cos(x) – \sin(x)\)

  11. What is the limit of \(\lim_{x \to -2} \frac{x^2 + 4x}{x + 2}\)?

    (a) 0  (b) 1   (c) -1   (d) Undefined

  12. What is the limit of \(\lim_{x \to 2} \frac{x^3 – 8}{x – 2}\)?

    (a) 10  (b) 12   (c) 5   (d) 21

  13. What is the limit of \(\lim_{x \to 0} \frac{\sin(2x)}{x}\)?

    (a) \(1/2\)  (b) 2   (c) -1   (d) 0

  14. Find the limit as x approaches infinity for the function \(f(x) = \frac{x^2 + 3x – 1}{2x^2 – x + 2}\)?

    (a) \(1/2\)  (b) 2   (c) \(\infty\)   (d) -1

  15. What is the limit of \(\lim_{x \to -2} \frac{x^2 + 4x+1}{x^2 + 2x}\)?

    (a) -1  (b) 0   (c) 1   (d) \(\infty\)

  16. What is the limit of \(\lim_{x \to 0} \frac{cosecx}{1/x}\)?

    (a) 0  (b) 1   (c) -1   (d) -2

  17. What is the limit of \(\lim_{x \to -1} \frac{x^3 + 1}{x^2 – 1}\)?

    (a) 0  (b) -1   (c)\(-3/2\)   (d) -2

  18. What is the limit of \(\lim_{x \to 4} \frac{\sqrt{x} – 2}{x – 4}\)?

    (a) 0  (b) \(1/4\)   (c) 4   (d) Undefined

  19. What is the limit of \(\lim_{x \to 0} \frac{ sin(5x)}{3x}\)?

    (a) 1  (b) \(3/5\)   (c) \(5/3\)   (d) 5

  20. What is the limit of \(\lim_{x \to 0} \frac{\tan(x)}{x}\)?

    (a) 0  (b) 1   (c) -1   (d) Undefined

Answers- (1) b (2) c (3) a (4) b (5) a (6) a  (7) b (8) b (9) a (10) a (11) a (12) b (13) b (14) a (15) d (16) b (17) c (18) b (19) c (20) b

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Last Update: July 23, 2024

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