Question

Let lim x→a g(x) = 1 , lim x→a f(x) = 0, lim x→a h(x) = 7. Find following limits if they exist. If not, enter DNE (’does not exist’) as your answer.
1. lim x→a (g(x) + f(x)) 2. lim x→a (g(x)− f(x)) 3. lim x→a (g(x) ∗ h(x)) 4. lim x→a g(x) f(x) 5. lim x→a g(x) h(x) 6. lim x→a h(x) g(x) 7. lim x→a p f(x) 8. lim x→a f(x) −1 9. lim x→a 1 f(x)−h(x)

Answer 1

Answer:

Using the properties of the limits, we have that:

1. $lim_{x\rightarrow a}(g(x)+f(x))=lim_{x\rightarrow a}g(x)+lim_{x\rightarrow a}f(x)=1+0=1$

2. $lim_{x\rightarrow a}(g(x)-f(x))=lim_{x\rightarrow a}g(x)-lim_{x\rightarrow a}f(x)=1-0=1$

3. $lim_{x\rightarrow a}(g(x)*h(x))=lim_{x\rightarrow a}g(x)*lim_{x\rightarrow a}h(x)=1*7=7$

4. $lim_{x\rightarrow a}(g(x)*f(x))=lim_{x\rightarrow a}g(x)*lim_{x\rightarrow a}f(x)=1*0=0$

5. $lim_{x\rightarrow a}(\frac{g(x)}{h(x)})=\frac{lim_{x\rightarrow a}g(x)}{lim_{x\rightarrow a}h(x)}=\frac{1}{7}$

6. $lim_{x\rightarrow a}(\frac{h(x)}{g(x)})=\frac{lim_{x\rightarrow a}h(x)}{lim_{x\rightarrow a}g(x)}=\frac{7}{1}=7$

7. $lim_{x\rightarrow a} pf(x)=plim_{x\rightarrow a}f(x)=p*0=0$

8. $lim_{x\rightarrow a}f(x)-1=lim_{x\rightarrow a}f(x)-lim_{x\rightarrow a}1=0-1=-1$

9. $lim_{x\rightarrow a}\frac{1}{f(x)-h(x)}=\frac{lim_{x\rightarrow a}1}{lim_{x\rightarrow a}f(x)-h(x)}=\frac{1}{lim_{x\rightarrow a}f(x)-lim_{x\rightarrow a}h(x)}=\frac{1}{0-7}=-\frac{1}{7}$