CBSE 11 Limits and Derivatives Practice Paper

Limits and Derivatives

1. Find the following limits

        (a) \(\lim\limits_{x\to 1}{\frac{x^{15}-1}{x^{10}-1}}\)                (b)\(\lim\limits_{z\to 0}\frac{\sqrt{1+z}-1}{z}\)

        (c) \(\lim\limits_{x\to 1}{[x^3-x^2+1]}\)       (d)  \(\lim\limits_{x\to 2}{[x(x+2)]}\)

        (e) \(\lim\limits_{x\to 1}{[\frac{x+1}{x+5}]}\)                (f)  \(\lim\limits_{x\to 1}{[\frac{x^3-2x^2+2x}{x^2-2}]}\)

2. For any positive integer n, prove that

    \(\lim\limits_{x\to a}{\frac{x^n-a^n}{x-a}}\)

3. Evaluate

        (a) \(\lim\limits_{x\to 0}{\frac{sin4x}{sin3x}}\)                (b) \(\lim\limits_{x\to 0}{\frac{sin{ax}}{sinbx}}\)

        (c) \(\lim\limits_{x\to 0}{\frac{\frac{1}{x}+\frac{1}{2}}{x+2}}\)                (d) \(\lim\limits_{x\to 0}{\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}}\)

4. Find \(\lim\limits_{x\to 0}{f(x)}\) and \(\lim\limits_{x\to 1}{f(x)}\), where\(f(x)=\begin{cases}2x+3,  x\leq0 \\[2ex] 3(x+1), x\gt 0\end{cases}\)

5. If  \(f(x)=\begin{cases}mx^{2}+n, x\lt 0 \\[2ex] nx+m, 0\leq x\geq 1. \\[2ex] nx^2+m,  x\gt1\end{cases}\) For what integers m and n does both \(\lim\limits_{x\to 0}{f(x)}\) and \(\lim\limits_{x\to 1}{f(x)}\) exist?

6. Find the derivative of  the following  functions 

 (a)\(f(x) = 10x\)            (b) \(f(x) = x^2\)

 (c)\(f(x) = {x+1}{x}\)        (d) \(f(x) = tanx\)

 (e)\(f(x) = cos x\)            (f) \(f(x) = 3cot x + 5cosec x\)

7. Find the derivative of f from the first principle, where f is given by

(a) \(f(x) = \frac{2x+3}{x-2}\)           (b) \(f(x) = x+\frac{1}{x}\)

(c) \(f(x) = sinx+cosx\)    (d) \(f(x) = sin{2x}\)

8. Find the derivative of the following functions (a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers)

(a) \(\frac{ax+b}{cx+d}\)     (c) \(\frac{ax+b}{px^2+qx+r}\)

(b) \(\frac{1+\frac{1}{x}}{1-\frac{1}{x}}\) (d) \((x+secx)(x-tanx)\)

(e) \(\frac{4x+5sinx}{3x+7cotx}\)    (f) \(\frac{ax^2+sinx}{p+qcosx}\)

(g) \(\frac{secx+1}{secx-1}\)     (h) \(\frac{1}{ax^2+bx+c}\)

9. Find the derivative of the function \(f(x) = 2x^2 + 3x – 5x at x =  –1\). Also prove that \(f'(0) + 3f'(-1)=0\).

10. Find the derivative at \(x = 2\) of the function \(f(x) = 3x\).

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