Evaluate \(\lim_{x\to 3} (x+3)\).
6
Evaluate \(\lim_{x\to \pi} (x - \tfrac{22}{7})\).
\(\pi - \tfrac{22}{7}\)
Evaluate \(\lim_{r\to 1} \pi r^{2}\).
\(\pi\)
Evaluate \(\lim_{x\to 4} \dfrac{4x+3}{x-2}\).
\(\tfrac{19}{2}\)
Evaluate \(\lim_{x\to 1} \dfrac{x^{10}+x^{5}+1}{x-1}\).
\(-\tfrac{1}{2}\)
Evaluate \(\lim_{x\to 0} \dfrac{(x+1)^{5}-1}{x}\).
5
Evaluate \(\lim_{x\to 2} \dfrac{3x^{2}-x-10}{x^{2}-4}\).
\(\tfrac{11}{4}\)
Evaluate \(\lim_{x\to 3} \dfrac{x^{4}-81}{2x^{2}-5x-3}\).
\(\tfrac{108}{7}\)
Evaluate \(\lim_{x\to 0} \dfrac{ax+b}{cx+1}\).
\(b\)
Evaluate \(\lim_{z\to 1} \dfrac{z^{1/3}-1}{z^{1/6}-1}\).
2
Evaluate \(\lim_{x\to 1} \dfrac{ax^{2}+bx+c}{cx^{2}+bx+a}\), where \(a+b+c\neq 0\).
1
Evaluate \(\lim_{x\to -2} \dfrac{\tfrac{1}{x}+\tfrac{1}{2}}{x+2}\).
\(-\tfrac{1}{4}\)
Evaluate \(\lim_{x\to 0} \dfrac{\sin ax}{bx}\).
\(\tfrac{a}{b}\)
Evaluate \(\lim_{x\to 0} \dfrac{\sin ax}{\sin bx}\), where \(a,b\neq 0\).
\(\tfrac{a}{b}\)
Evaluate \(\lim_{x\to \pi} \dfrac{\sin(\pi - x)}{\pi(\pi - x)}\).
\(\tfrac{1}{\pi}\)
Evaluate \(\lim_{x\to 0} \dfrac{\cos x}{\pi - x}\).
\(\tfrac{1}{\pi}\)
Evaluate \(\lim_{x\to 0} \dfrac{\cos 2x - 1}{\cos x - 1}\).
4
Evaluate \(\lim_{x\to 0} \dfrac{ax + x\cos x}{b\sin x}\).
\(\tfrac{a+1}{b}\)
Evaluate \(\lim_{x\to 0} x\sec x\).
0
Evaluate \(\lim_{x\to 0} \dfrac{\sin ax + bx}{ax + \sin bx}\), where \(a, b, a+b \neq 0\).
1
Evaluate \(\lim_{x\to 0} (\csc x - \cot x)\).
0
Evaluate \(\lim_{x\to \pi/2} \dfrac{\tan 2x}{x - \pi/2}\).
2
Find \(\lim_{x\to 0} f(x)\) and \(\lim_{x\to 1} f(x)\), where
\(f(x) = \begin{cases} 2x+3, & x \le 0 \\ 3(x+1), & x>0 \end{cases}\).
\(\lim_{x\to 0} f(x) = 3\)
\(\lim_{x\to 1} f(x) = 6\)
Find \(\lim_{x\to 1} f(x)\), where
\(f(x) = \begin{cases} x^{2}-1, & x \le 1 \\ -x^{2}-1, & x>1 \end{cases}\).
Limit does not exist at \(x=1\).
Evaluate \(\lim_{x\to 0} f(x)\), where
\(f(x) = \begin{cases} \dfrac{|x|}{x}, & x \ne 0 \\ 0, & x=0 \end{cases}\).
Limit does not exist at \(x=0\).
Find \(\lim_{x\to 0} f(x)\), where
\(f(x) = \begin{cases} \dfrac{x}{|x|}, & x \ne 0 \\ 0, & x=0 \end{cases}\).
Limit does not exist at \(x=0\).
Find \(\lim_{x\to 5} f(x)\), where \(f(x) = |x| - 5\).
0
Suppose
\(f(x) = \begin{cases} a+bx, & x<1 \\ 4, & x=1 \\ b-ax, & x>1 \end{cases}\)
and if \(\lim_{x\to 1} f(x) = f(1)\), what are possible values of \(a\) and \(b\)?
\(a=0,\ b=4\)
Let \(a_{1}, a_{2}, \ldots, a_{n}\) be fixed real numbers and define
\(f(x) = (x-a_{1})(x-a_{2})\cdots(x-a_{n})\).
What is \(\lim_{x\to a_{1}} f(x)\)? For some \(a \ne a_{1}, a_{2}, \ldots, a_{n}\), compute \(\lim_{x\to a} f(x)\).
\(\lim_{x\to a_{1}} f(x) = 0\)
\(\lim_{x\to a} f(x) = (a-a_{1})(a-a_{2})\cdots(a-a_{n})\)
If
\(f(x) = \begin{cases} |x|+1, & x<0 \\ 0, & x=0 \\ |x|-1, & x>0 \end{cases}\),
for what value(s) of \(a\) does \(\lim_{x\to a} f(x)\) exist?
\(\lim_{x\to a} f(x)\) exists for all \(a \ne 0\).
If the function \(f(x)\) satisfies
\(\displaystyle \lim_{x\to 1} \dfrac{f(x) - 2}{x^{2} - 1} = \pi\),
evaluate \(\lim_{x\to 1} f(x)\).
2
If
\(f(x) = \begin{cases} mx^{2}+n, & x<0 \\ nx+m, & 0 \le x \le 1 \\ nx^{3}+m, & x>1 \end{cases}\).
For what integers \(m\) and \(n\) does both \(\lim_{x\to 0} f(x)\) and \(\lim_{x\to 1} f(x)\) exist?
For \(\lim_{x\to 0} f(x)\) to exist, we need \(m=n\); \(\lim_{x\to 1} f(x)\) exists for any integral values of \(m\) and \(n\).
Find the derivative of \(x^{2}-2\) at \(x=10\).
20
Find the derivative of \(x\) at \(x=1\).
1
Find the derivative of \(99x\) at \(x=100\).
99
Find the derivative of the following functions from first principle:
(i) \(x^{3} - 27\)
(ii) \((x-1)(x-2)\)
(iii) \(\dfrac{1}{x^{2}}\)
(iv) \(\dfrac{x+1}{x-1}\)
(i) \(3x^{2}\)
(ii) \(2x - 3\)
(iii) \(-\dfrac{2}{x^{3}}\)
(iv) \(-\dfrac{2}{(x-1)^{2}}\)
For the function
\(f(x)=\dfrac{x^{100}}{100}+\dfrac{x^{99}}{99}+\cdots+\dfrac{x^{2}}{2}+x+1\),
prove that \(f'(1)=100f'(0)\).
Result: \(f'(1)=100f'(0)\)
Find the derivative of the expression:
\(x^{n}+ax^{n-1}+a^{2}x^{n-2}+\cdots +a^{n-1}x+a^{n}\)
\(nx^{n-1}+a(n-1)x^{n-2}+a^{2}(n-2)x^{n-3}+\cdots +a^{n-1}\)
For some constants \(a\) and \(b\), find the derivative of the following:
(i) \((x-a)(x-b)\)
(ii) \((ax^{2}+b)^{2}\)
(iii) \(\dfrac{x-a}{x-b}\)
(i) \(2x - a - b\)
(ii) \(4ax(ax^{2}+b)\)
(iii) \(\dfrac{a-b}{(x-b)^{2}}\)
Find the derivative of
\(\dfrac{x^{n}-a^{n}}{x-a}\)
for constant \(a\).
\(\dfrac{nx^{n}-anx^{n-1}-x^{n}+a^{n}}{(x-a)^{2}}\)
Find the derivative of the following:
(i) \(2x - \dfrac{3}{4}\)
(ii) \((5x^{3}+3x-1)(x-1)\)
(iii) \(x^{-3}(5+3x)\)
(iv) \(x^{5}(3-6x^{-9})\)
(v) \(x^{-4}(3-4x^{-5})\)
(vi) \(\dfrac{2}{x+1} - \dfrac{x^{2}}{3x-1}\)
(i) 2
(ii) \(20x^{3} - 15x^{2} + 6x - 4\)
(iii) \(-\dfrac{3}{x^{4}}(5+2x)\)
(iv) \(15x^{4} + \dfrac{24}{x^{5}}\)
(v) \(-\dfrac{12}{x^{5}} + \dfrac{36}{x^{10}}\)
(vi) \(-\dfrac{2}{(x+1)^{2}} - \dfrac{x(3x-2)}{(3x-1)^{2}}\)
Find the derivative of \(\cos x\) from first principle.
-\sin x
Find the derivative of the following functions:
(i) \(\sin x \cos x\)
(ii) \(\sec x\)
(iii) \(5\sec x + 4\cos x\)
(iv) \(\csc x\)
(v) \(3\cot x + 5\cosec x\)
(vi) \(5\sin x - 6\cos x + 7\)
(vii) \(2\tan x - 7\sec x\)
(i) \(\cos 2x\)
(ii) \(\sec x \tan x\)
(iii) \(5\sec x \tan x - 4\sin x\)
(iv) \(-\csc x \cot x\)
(v) \(-3\csc^{2}x - 5\csc x \cot x\)
(vi) \(5\cos x + 6\sin x\)
(vii) \(2\sec^{2}x - 7\sec x \tan x\)
Find the derivative of the following functions from first principle:
(i) \(-x\)
(ii) \((-x)^{-1}\)
(iii) \(\sin (x+1)\)
(iv) \(\cos \left(x-\tfrac{\pi}{8}\right)\)
(i) \(-1\)
(ii) \(\dfrac{1}{x^{2}}\)
(iii) \(\cos (x+1)\)
(iv) \(-\sin \left(x-\tfrac{\pi}{8}\right)\)
Find the derivative of \(x+a\).
1
Find the derivative of \((px+q)\left(\dfrac{r}{x}+s\right)\), where \(p,q,r,s\) are fixed non-zero constants.
\(-\dfrac{qr}{x^{2}}+ps\)
Find the derivative of \((ax+b)(cx+d)^{2}\).
\(2c(ax+b)(cx+d)+a(cx+d)^{2}\)
Find the derivative of \(\dfrac{ax+b}{cx+d}\).
\(\dfrac{ad-bc}{(cx+d)^{2}}\)
Find the derivative of \(\dfrac{1+\tfrac{1}{x}}{1-\tfrac{1}{x}}\).
\(-\dfrac{2}{(x-1)^{2}},\ x\neq 0,1\)
Find the derivative of \(\dfrac{1}{ax^{2}+bx+c}\).
\(-\dfrac{2ax+b}{(ax^{2}+bx+c)^{2}}\)
Find the derivative of \(\dfrac{ax+b}{px^{2}+qx+r}\).
\(-\dfrac{apx^{2}+2bpx+ar-bq}{(px^{2}+qx+r)^{2}}\)
Find the derivative of \(\dfrac{px^{2}+qx+r}{ax+b}\).
\(\dfrac{apx^{2}+2bpx+bq-ar}{(ax+b)^{2}}\)
Find the derivative of \(\dfrac{a}{x^{4}}-\dfrac{b}{x^{2}}+\cos x\).
\(-\dfrac{4a}{x^{5}}+\dfrac{2b}{x^{3}}-\sin x\)
Find the derivative of \(4\sqrt{x}-2\).
\(\dfrac{2}{\sqrt{x}}\)
Find the derivative of \((ax+b)^{n}\).
\(na(ax+b)^{n-1}\)
Find the derivative of \((ax+b)^{n}(cx+d)^{m}\).
\((ax+b)^{n-1}(cx+d)^{m-1}\big[mc(ax+b)+na(cx+d)\big]\)
Find the derivative of \(\sin(x+a)\).
\(\cos(x+a)\)
Find the derivative of \(\csc x\cot x\).
\(-\csc^{3}x-\csc x\cot^{2}x\)
Find the derivative of \(\dfrac{\cos x}{1+\sin x}\).
\(-\dfrac{1}{1+\sin x}\)
Find the derivative of \(\dfrac{\sin x+\cos x}{\sin x-\cos x}\).
\(-\dfrac{2}{(\sin x-\cos x)^{2}}\)
Find the derivative of \(\dfrac{\sec x-1}{\sec x+1}\).
\(\dfrac{2\sec x\tan x}{(\sec x+1)^{2}}\)
Find the derivative of \(\sin^{n}x\).
\(n\sin^{n-1}x\cos x\)
Find the derivative of \(\dfrac{a+b\sin x}{c+d\cos x}\).
\(\dfrac{bc\cos x+ad\sin x+bd}{(c+d\cos x)^{2}}\)
Find the derivative of \(\dfrac{\sin(x+a)}{\cos x}\).
\(\dfrac{\cos a}{\cos^{2}x}\)
Find the derivative of \(x^{4}(5\sin x-3\cos x)\).
\(x^{3}\big(5x\cos x+3x\sin x+20\sin x-12\cos x\big)\)
Find the derivative of \((x^{2}+1)\cos x\).
\(-x^{2}\sin x-\sin x+2x\cos x\)
Find the derivative of \((ax^{2}+\sin x)(p+q\cos x)\).
\(-q\sin x(ax^{2}+\sin x)+(p+q\cos x)(2ax+\cos x)\)
Find the derivative of \((x+\cos x)(x-\tan x)\).
\(-\tan^{2}x(x+\cos x)+(x-\tan x)(1-\sin x)\)
Find the derivative of \(\dfrac{4x+5\sin x}{3x+7\cos x}\).
\(\dfrac{35+15x\cos x+28\cos x+28x\sin x-15\sin x}{(3x+7\cos x)^{2}}\)
Find the derivative of \(\dfrac{x^{2}\cos\left(\tfrac{\pi}{4}\right)}{\sin x}\).
\(\dfrac{x\cos\left(\tfrac{\pi}{4}\right)(2\sin x-x\cos x)}{\sin^{2}x}\)
Find the derivative of \(\dfrac{x}{1+\tan x}\).
\(\dfrac{1+\tan x-x\sec^{2}x}{(1+\tan x)^{2}}\)
Find the derivative of \((x+\sec x)(x-\tan x)\).
\((x+\sec x)(1-\sec^{2}x)+(x-\tan x)(1+\sec x\tan x)\)
Find the derivative of \(\dfrac{x}{\sin^{n}x}\).
\(\dfrac{\sin x-nx\cos x}{\sin^{n+1}x}\)