Evaluate \(\displaystyle\lim_{x\to3}\dfrac{x^{2}-9}{x-3}\).
6
Evaluate \(\displaystyle\lim_{x\to\tfrac{1}{2}}\dfrac{4x^{2}-1}{2x-1}\).
2
Evaluate \(\displaystyle\lim_{h\to0}\dfrac{\sqrt{x+h}-\sqrt{x}}{h}\).
\(\dfrac{1}{2\sqrt{x}}\)
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{(x+2)^{1/3}-2^{1/3}}{x}\).
\(\dfrac{1}{3}2^{-2/3}\)
Evaluate \(\displaystyle\lim_{x\to1}\dfrac{(1+x)^{6}-1}{(1+x)^{2}-1}\).
3
Evaluate \(\displaystyle\lim_{x\to-1}\dfrac{x^{3}+27}{x+1}\).
3
Evaluate \(\displaystyle\lim_{x\to1}\dfrac{x^{4}-\sqrt{x}}{\sqrt{x}-1}\).
7
Evaluate \(\displaystyle\lim_{x\to2}\dfrac{x^{2}-4}{\sqrt{3x-2}-\sqrt{x+2}}\).
8
Evaluate \(\displaystyle\lim_{x\to\sqrt{2}}\dfrac{x^{4}-4}{x^{2}+3\sqrt{2}x-8}\).
\(\dfrac{8}{5}\)
Evaluate \(\displaystyle\lim_{x\to1}\dfrac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}\).
1
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sqrt{1+x^{3}}-\sqrt{1-x^{3}}}{x^{2}}\).
0
Evaluate \(\displaystyle\lim_{x\to-3}\dfrac{x^{3}+27}{x^{5}+243}\).
\(\dfrac{1}{15}\)
Evaluate \(\displaystyle\lim_{x\to\tfrac{1}{2}}\left(\dfrac{8x-3}{2x-1}-\dfrac{4x^{2}+1}{4x^{2}-1}\right)\).
\(\dfrac{7}{2}\)
Find \(n\in\mathbb{N}\) if \(\displaystyle\lim_{x\to2}\dfrac{x^{n}-2^{n}}{x-2}=80\).
\(n=5\)
Evaluate \(\displaystyle\lim_{x\to a}\dfrac{\sin3x}{\sin7x}\).
\(\dfrac{3}{7}\)
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sin^{2}2x}{\sin^{2}4x}\).
\(\dfrac{1}{4}\)
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{1-\cos2x}{x^{2}}\).
2
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{2\sin x-\sin2x}{x^{3}}\).
1
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{1-\cos mx}{1-\cos nx}\).
\(\dfrac{m^{2}}{n^{2}}\)
Evaluate \(\displaystyle\lim_{x\to\tfrac{\pi}{3}}\sqrt{2}\left(\dfrac{\pi}{3}-x\right)\sqrt{1-\cos6x}\).
3
Evaluate \(\displaystyle\lim_{x\to\tfrac{\pi}{4}}\dfrac{\sin x-\cos x}{x-\tfrac{\pi}{4}}\).
\(\sqrt{2}\)
Evaluate \(\displaystyle\lim_{x\to\tfrac{\pi}{6}}\dfrac{\sqrt{3}\sin x-\cos x}{x-\tfrac{\pi}{6}}\).
2
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sin2x+3x}{2x+\tan3x}\).
1
Evaluate \(\displaystyle\lim_{x\to a}\dfrac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}\).
\(2\sqrt{a}\cos a\)
Evaluate \(\displaystyle\lim_{x\to\tfrac{\pi}{6}}\dfrac{\cot^{2}x-3}{\cosec x-2}\).
4
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sqrt{2}-\sqrt{1+\cos x}}{\sin^{2}x}\).
\(\dfrac{1}{4\sqrt{2}}\)
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sin x-2\sin3x+\sin5x}{x}\).
0
If \(\displaystyle\lim_{x\to1}\dfrac{x^{4}-1}{x-1}=\lim_{x\to k}\dfrac{x^{3}-k^{3}}{x^{2}-k^{2}}\), then find \(k\).
\(k=\dfrac{3}{8}\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{x^{4}+x^{3}+x^{2}+1}{x}\).
\(3x^{2}+2x+1-\dfrac{1}{x^{2}}\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=\left(x+\dfrac{1}{x}\right)^{3}\).
\(3\left(x+\dfrac{1}{x}\right)^{2}\left(1-\dfrac{1}{x^{2}}\right)\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=(3x+5)(1+\tan x)\).
\(3(1+\tan x)+(3x+5)\sec^{2}x\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=(\sec x-1)(\sec x+1)=\sec^{2}x-1\).
\(2\sec^{2}x\tan x\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{3x+4}{5x^{2}-7x+9}\).
\(\dfrac{55-40x-15x^{2}}{(5x^{2}-7x+9)^{2}}\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{x^{5}-\cos x}{\sin x}\).
\(\dfrac{(5x^{4}+\sin x)\sin x-(x^{5}-\cos x)\cos x}{\sin^{2}x}\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{x^{2}\cos\tfrac{\pi}{4}}{\sin x}\) (note \(\cos\tfrac{\pi}{4}=\tfrac{1}{\sqrt{2}}\)).
\(\dfrac{\sqrt{2}\,x\sin x- x^{2}\cos x}{2\sin^{2}x}\) (equivalently, compute using quotient rule with constant \(\cos\tfrac{\pi}{4}\))
Differentiate with respect to \(x\): \(\displaystyle f(x)=(ax^{2}+\cot x)(p+q\cos x)\).
\( (2ax-\csc^{2}x)(p+q\cos x)+(ax^{2}+\cot x)(-q\sin x)\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{a+b\sin x}{c+d\cos x}\).
\(\dfrac{(b\cos x)(c+d\cos x)-(a+b\sin x)(-d\sin x)}{(c+d\cos x)^{2}}\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=(\sin x+\cos x)^{2}\).
\(2(\sin x+\cos x)(\cos x-\sin x)\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=(2x-7)^{2}(3x+5)^{3}\).
Use product rule: \(2(2x-7)(2)(3x+5)^{3}+(2x-7)^{2}\cdot3(3x+5)^{2}\cdot3\).
Differentiate with respect to \(x\): \(\displaystyle f(x)=x^{2}\sin x+2x\sin x-2\sin2x\).
\(2x\sin x+x^{2}\cos x+2\sin x+2x\cos x-4\cos2x\)
Differentiate with respect to \(x\): \(\displaystyle f(x)=\sin^{3}x\cos^{3}x\).
\(3\sin^{2}x\cos^{3}x\cos x+3\sin^{3}x\cos^{2}x(-\sin x)\) (apply product rule or write \( (\sin x\cos x)^{3}\)).
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{1}{ax^{2}+bx+c}\).
\(-\dfrac{2ax+b}{(ax^{2}+bx+c)^{2}}\)
Differentiate with respect to x: \(\cos\big(x^{2}+1\big)\).
\(-2x\sin\big(x^{2}+1\big)\)
Differentiate with respect to x: \(\dfrac{ax+b}{cx+d}\).
\(\dfrac{ad-bc}{(cx+d)^{2}}\)
Differentiate with respect to x: \(x^{2/3}\).
\(\dfrac{2}{3}x^{-1/3}\)
Differentiate with respect to x: \(x\cos x\).
\(\cos x - x\sin x\)
Evaluate: \(\displaystyle \lim_{y\to 0} \dfrac{(x+y)\sec(x+y)-x\sec x}{y}\).
\(\sec x\big(x\tan x+1\big)\)
Evaluate: \(\displaystyle \lim_{x\to 0} \dfrac{\sin(\alpha+\beta)x+\sin(\alpha-\beta)x+\sin 2\alpha x}{\cos 2\beta x-\cos 2\alpha x}\cdot x\).
\(\dfrac{2\alpha}{\alpha^{2}-\beta^{2}}\)
Evaluate: \(\displaystyle \lim_{x\to \tfrac{\pi}{4}} \dfrac{\tan^{3}x-\tan x}{\cos\big(x+\tfrac{\pi}{4}\big)}\).
-4
Evaluate: \(\displaystyle \lim_{x\to\pi} \dfrac{1-\sin\tfrac{x}{2}}{\cos\tfrac{x}{2}\big(\cos\tfrac{x}{4}-\sin\tfrac{x}{4}\big)}\).
\(\dfrac{1}{\sqrt{2}}\)
Evaluate (show whether limit exists): \(\displaystyle \lim_{x\to 4} \dfrac{|x-4|}{x-4}\).
Does not exist
Evaluate (find constant): Let \(f(x)=\begin{cases}\dfrac{k\cos x}{\pi-2x}&\text{when }x\neq\dfrac{\pi}{2},\\[6pt]3&\text{when }x=\dfrac{\pi}{2}.\end{cases}\) If \(\displaystyle\lim_{x\to\tfrac{\pi}{2}} f(x)=f\big(\tfrac{\pi}{2}\big)\), find \(k\).
\(6\)
Evaluate (find constant): Let \(f(x)=\begin{cases}x+2,&x\le 1,\\[6pt]cx^{2},&x>-1.\end{cases}\) Find \(c\) if \(\displaystyle\lim_{x\to -1} f(x)\) exists.
\(1\)
\(\lim_{x\to\pi} \dfrac{\sin x}{x-\pi}\) is
1
2
-1
-2
\(\lim_{x\to 0} \dfrac{x^{2}\cos x}{1-\cos x}\) is
2
\(\tfrac{3}{2}\)
\(-\tfrac{3}{2}\)
1
\(\lim_{x\to 0} \dfrac{(1+x)^{n}-1}{x}\) is
\(n\)
1
\(-n\)
0
\(\lim_{x\to 1} \dfrac{x^{m}-1}{x^{n}-1}\) is
1
\(\dfrac{m}{n}\)
\(-\dfrac{m}{n}\)
\(\dfrac{m^{2}}{n^{2}}\)
\(\lim_{\theta\to 0} \dfrac{1-\cos 4\theta}{1-\cos 6\theta}\) is
\(\dfrac{4}{9}\)
\(\dfrac{1}{2}\)
\(-\dfrac{1}{2}\)
-1
\(\lim_{x\to 0} \dfrac{\csc x - \cot x}{x}\) is
\(-\tfrac{1}{2}\)
1
\(\tfrac{1}{2}\)
1
\(\lim_{x\to 0} \dfrac{\sin x}{\sqrt{x+1}-\sqrt{1-x}}\) is
2
0
1
-1
\(\lim_{x\to \dfrac{\pi}{4}} \dfrac{\sec^{2}x -2}{\tan x -1}\) is
3
1
0
\(\sqrt{2}\)
\(\lim_{x\to 1} \dfrac{(\sqrt{x}-1)(2x-3)}{2x^{2}+x-3}\) is
\(\dfrac{1}{10}\)
\(-\dfrac{1}{10}\)
1
None of these
Let \(f(x)=\begin{cases}\dfrac{\sin[ x ]}{[ x ]},&[ x ]\ne 0\\0,&[ x ]=0\end{cases}\) where \([\cdot]\) denotes the greatest integer function. Then \(\lim_{x\to 0} f(x)\) is
1
0
-1
None of these
\(\lim_{x\to 0} \dfrac{|\sin x|}{x}\) is
1
-1
does not exist
None of these
Let \(f(x)=\begin{cases}x^{2}-1,&0
\(x^{2}-6x+9=0\)
\(x^{2}-7x+8=0\)
\(x^{2}-14x+49=0\)
\(x^{2}-10x+21=0\)
\(\lim_{x\to 0} \dfrac{\tan 2x - x}{3x - \sin x}\)
2
\(\dfrac{1}{2}\)
\(-\dfrac{1}{2}\)
\(\dfrac{1}{4}\)
Let \(f(x)=x-[x]\), \(x\in\mathbb{R}\). Then \(f'(\tfrac{1}{2})\) is
\(\dfrac{3}{2}\)
1
0
-1
If \(y=\sqrt{x}+\dfrac{1}{\sqrt{x}}\), then \(\dfrac{dy}{dx}\) at \(x=1\) is
1
\(\dfrac{1}{2}\)
\(\dfrac{1}{\sqrt{2}}\)
0
If \(f(x)=\dfrac{x-4}{2\sqrt{x}}\), then \(f'(1)\) is
\(\dfrac{5}{4}\)
\(\dfrac{4}{5}\)
1
0
If \(y=\dfrac{1+\dfrac{1}{x^{2}}}{1-\dfrac{1}{x^{2}}}\), then \(\dfrac{dy}{dx}\) is
\(-\dfrac{4x}{(x^{2}-1)^{2}}\)
\(-\dfrac{4x}{x^{2}-1}\)
\(\dfrac{1-x^{2}}{4x}\)
\(\dfrac{4x}{x^{2}-1}\)
If \(y=\dfrac{\sin x+\cos x}{\sin x-\cos x}\), then \(\dfrac{dy}{dx}\) at \(x=0\) is
-2
0
\(\dfrac{1}{2}\)
does not exist
If \(y=\dfrac{\sin(x+9)}{\cos x}\), then \(\dfrac{dy}{dx}\) at \(x=0\) is
\(\cos 9\)
\(\sin 9\)
0
1
If \(f(x)=1+x+\dfrac{x^{2}}{2}+\dots+\dfrac{x^{100}}{100}\), then \(f'(1)\) is equal to
\(\dfrac{1}{100}\)
100
does not exist
0
If \(f(x)=\dfrac{x^{n}-a^{n}}{x-a}\) for some constant \(a\), then \(f'(a)\) is
1
0
does not exist
\(\dfrac{1}{2}\)
If \(f(x)=x^{100}+x^{99}+\dots+x+1\), then \(f'(1)\) is equal to
5050
5049
5051
50051
If \(f(x)=1-x+x^{2}-x^{3}+\dots+ x^{100}\), then \(f'(1)\) is equal to
150
-50
-150
50
If \( f(x) = \dfrac{\tan x}{x - \pi} \), then \( \lim_{x \to \pi} f(x) = \_\_\_\_\_\_\_\_\_\_ \)
1
\( \lim_{x \to 0} \left( \sin(mx) \cot \dfrac{x}{\sqrt{3}} \right) = 2 \), then \( m = \_\_\_\_\_\_\_\_ \)
\( \dfrac{2\sqrt{3}}{3} \)
If \( y = 1 + \dfrac{x}{1!} + \dfrac{x^{2}}{2!} + \dfrac{x^{3}}{3!} + \ldots \), then \( \dfrac{dy}{dx} = \_\_\_\_\_\_\_\_ \)
y
\( \lim_{x \to 3^{+}} \dfrac{x}{[x]} = \_\_\_\_\_\_\_ \)
1