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- Homework Statement
- Suppose Lim x->c f(x)=L1 and Lim x->c g(x)=L2. Define h by h(x)=f(x)g(x). Prove that Lim x->c h(x)=L1L2

- Relevant Equations
- 0< |x-c| <δ -> |f(x)-L1|<ϵ1 and |g(x)-L2|<ϵ2

When I look at a range of inputs around x=c and consider the corresponding range of outputs

If 0< |x-c| <δ -> |f(x)-L1|<ϵ1 and |g(x)-L2|<ϵ2 as we shrink the range of inputs the corresponding outputs f(x) and g(x) narrow on L1 and L2 respectively.

|f(x)-L1||g(x)-L2|<ϵ2ϵ1

The product of the absolute values is equal to the absolute value of the product.

|f(x)g(x)-f(x)L2-g(x)L1+L1L2|<ϵ2ϵ1

By definition h(x)=f(x)g(x)

|h(x)-f(x)L2-g(x)L1+L1L2|

Add and subtract L1L2

|(h(x)-L1L2)-f(x)L2-g(x)L1+2L1L2|

|(h(x)-L1L2)+L2(L1-f(x))+L1(L2-g(x))|<ϵ2ϵ1

The absolute value of a sum is less than or equal to the sum of their absolute values.

|(h(x)-L1L2)+L2(L1-f(x))+L1(L2-g(x))|<|(h(x)-L1L2)|+|L2||(L1-f(x))|+|L1||(L2-g(x))||<|(h(x)-L1L2)|+|L2|ϵ1+|L1|ϵ2

For |(h(x)-L1L2)|+|L2|ϵ1+|L1|ϵ2 to be less than ϵ2ϵ1

|(h(x)-L1L2)|<= ϵ2ϵ1-(|L2|ϵ1+|L1|ϵ2)

(|L1|/ϵ1)+(|L2|/ϵ2)<1

Does this imply that the epsilons must be greater than the limits to guarantee that the sum of the quotients be less than 1

If 0< |x-c| <δ -> |f(x)-L1|<ϵ1 and |g(x)-L2|<ϵ2 as we shrink the range of inputs the corresponding outputs f(x) and g(x) narrow on L1 and L2 respectively.

|f(x)-L1||g(x)-L2|<ϵ2ϵ1

The product of the absolute values is equal to the absolute value of the product.

|f(x)g(x)-f(x)L2-g(x)L1+L1L2|<ϵ2ϵ1

By definition h(x)=f(x)g(x)

|h(x)-f(x)L2-g(x)L1+L1L2|

Add and subtract L1L2

|(h(x)-L1L2)-f(x)L2-g(x)L1+2L1L2|

|(h(x)-L1L2)+L2(L1-f(x))+L1(L2-g(x))|<ϵ2ϵ1

The absolute value of a sum is less than or equal to the sum of their absolute values.

|(h(x)-L1L2)+L2(L1-f(x))+L1(L2-g(x))|<|(h(x)-L1L2)|+|L2||(L1-f(x))|+|L1||(L2-g(x))||<|(h(x)-L1L2)|+|L2|ϵ1+|L1|ϵ2

For |(h(x)-L1L2)|+|L2|ϵ1+|L1|ϵ2 to be less than ϵ2ϵ1

|(h(x)-L1L2)|<= ϵ2ϵ1-(|L2|ϵ1+|L1|ϵ2)

(|L1|/ϵ1)+(|L2|/ϵ2)<1

Does this imply that the epsilons must be greater than the limits to guarantee that the sum of the quotients be less than 1

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