Calculus

Limits of Functions

Limits of Functions: Level 2 Challenges

           

limx0+xx=?\large \displaystyle \lim_{x\to 0^+} x^x =\, ?

0.50.5=0.70710.40.4=0.69310.30.3=0.69680.20.2=0.72470.10.1=0.7943\begin{aligned} 0.5^{0.5} &=& 0.7071\ldots \\ 0.4^{0.4} &=& 0.6931\ldots \\ 0.3^{0.3} &=& 0.6968\ldots \\ 0.2^{0.2} &=& 0.7247\ldots \\ 0.1^{0.1} &=& 0.7943\ldots \\ \end{aligned}

limxsinxx=? \lim_{x\to\infty} \dfrac{\sin x}x = \, ?

limx02x12x+1x= ?\Large \lim_{x\rightarrow 0} \frac{|2x-1|-|2x+1|}{x}= \ ?

f(x)={(x24)/(x2), if x<22, if x=2x33x2+2x+4, if x>2f(x) = \large{\begin{cases} (x^2 - 4)/(x-2), & \text{ if } x < 2 \\ 2, & \text{ if } x = 2 \\ x^3-3x^2 + 2x + 4 , & \text{ if } x > 2 \\ \end{cases} }

Compute limx2f(x)\displaystyle \lim_{x \to 2} f(x).

Given that f(x)=1x1f(x) = \dfrac1{x-1} and g(x)=3x23x+2g(x) = \dfrac3{x^2-3x+2} , find limx1f(x)g(x) \displaystyle \lim_{x\to1} \dfrac{f(x)}{g(x)} .

limx0(sinx)(tanx)x2= ? \large \lim_{x \to 0} \, \left \lfloor \dfrac{(\sin x) (\tan x)}{x^2} \right \rfloor = \ ?

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