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

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)((g(x))/(h(x)))=(lim_(x\rightarrow a)g(x))/(lim_(x\rightarrow a)h(x))=(1)/(7)`

6.
`lim_(x\rightarrow a)((h(x))/(g(x)))=(lim_(x\rightarrow a)h(x))/(lim_(x\rightarrow a)g(x))=(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)(1)/(f(x)-h(x))=(lim_(x\rightarrow a)1)/(lim_(x\rightarrow a)f(x)-h(x))=(1)/(lim_(x\rightarrow a)f(x)-lim_(x\rightarrow a)h(x))=(1)/(0-7)=-(1)/(7)`