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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