Answer
a) $-\sqrt2$
b) $\sqrt2/2$
c) $\frac{1+2\sqrt2}{2}$
d) $2$
e) $1/2$
f) $-1/2$
Work Step by Step
$\lim_{x\to 0}f(x)=1/2$ and $\lim_{x\to 0}g(x)=\sqrt2$
a) $\lim_{x\to 0}(-g(x))=-(\lim_{x\to 0}g(x))=-\sqrt2$ (Composite Rule)
b) $\lim_{x\to 0}(g(x)\times f(x))=\lim_{x\to 0}g(x)\times\lim_{x\to 0}f(x)=\sqrt2\times(1/2)=\sqrt2/2$ (Product Rule)
c) $\lim_{x\to 0}(f(x)+g(x))=\lim_{x\to 0}f(x)+\lim_{x\to 0}g(x)=\frac{1}{2}+\sqrt2=\frac{1+2\sqrt2}{2}$ (Sum Rule)
d) $\lim_{x\to 0}1/f(x)=\frac{1}{\lim_{x\to 0}f(x)}=\frac{1}{1/2}=2$ (Quotient Rule)
e) $\lim_{x\to 0}(x+f(x))=\lim_{x\to 0}x+\lim_{x\to 0}f(x)=0+1/2=1/2$ (Sum Rule)
f) $\lim_{x\to 0}\frac{f(x)\cos x}{x-1}=\frac{\lim_{x\to 0}f(x)\times\lim_{x\to 0}(\cos x)}{\lim_{x\to 0}x-\lim_{x\to 0}1}=\frac{1/2\times\cos0}{0-1}=\frac{1/2\times1}{-1}=-1/2$ (Quotient, Product and Difference Rule)
