Determine whether the following functions are differentiable
i) 𝑓 (𝑥 )= |𝑥|;
ii) 𝑔(𝑥) = |𝑥| +| 𝑥 + 1 |
iii) h(x) = x^1/3
i:limx→0+f(x)−f(0)x−0=limx→0+xx=1limx→0−f(x)−f(0)x−0=limx→0+−xx=−1Not differentiableii:limx→0+g(x)−g(0)x−0=limx→0+x+1−1x=1limx→0−g(x)−g(0)x−0=limx→0+−x+1−1x=−1Not differentiableiii:limx→0h(x)−h(0)x−0=limx→0+x1/3x=∞Not differentiablei:\\\underset{x\rightarrow 0+}{\lim}\frac{f\left( x \right) -f\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{x}{x}=1\\\underset{x\rightarrow 0-}{\lim}\frac{f\left( x \right) -f\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{-x}{x}=-1\\Not\,\,differentiable\\ii:\\\underset{x\rightarrow 0+}{\lim}\frac{g\left( x \right) -g\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{x+1-1}{x}=1\\\underset{x\rightarrow 0-}{\lim}\frac{g\left( x \right) -g\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{-x+1-1}{x}=-1\\Not\,\,differentiable\\iii:\\\underset{x\rightarrow 0}{\lim}\frac{h\left( x \right) -h\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{x^{1/3}}{x}=\infty \\Not\,\,differentiablei:x→0+limx−0f(x)−f(0)=x→0+limxx=1x→0−limx−0f(x)−f(0)=x→0+limx−x=−1Notdifferentiableii:x→0+limx−0g(x)−g(0)=x→0+limxx+1−1=1x→0−limx−0g(x)−g(0)=x→0+limx−x+1−1=−1Notdifferentiableiii:x→0limx−0h(x)−h(0)=x→0+limxx1/3=∞Notdifferentiable
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