Question

If the limit of f(x) as x approaches a is infinity and the limit of g(x) as x approaches a is c, prove that the limit of [f(x)+g(x)] as x approaches a is infinity, when c>0.

Answer 1

hmm what do you need to prove exactly? because this seems self evident

Answer 2

They want us to use the defenition of positive infinite limit

Answer 3

\[\lim_{x\rightarrow a}f(x)=\infty\] translates to given any
\[N \hspace{.2cm}\exists \delta \text{ such that if }|x-a|<\delta, f(x)>N \]

Answer 4

since this is true for any N it is also true for N + c

Answer 5

Oops srry its supposed to be times

Answer 6

fine then replace N by cN still works

Answer 7

write the definition of the limit being infinity, replace N by cN since it is arbitrary in any case and you should be in good shape

Answer 8

Okay thanks

Answer 9

Limits can split up in this manner: \[\lim_{x\rightarrow \infty}[f(x)+g(x)]=\lim_{x\rightarrow \infty} f(x) + \lim_{x\rightarrow \infty} g(x)\]which, handwavingly, can be represented as follows: \[\infty + c = \infty\text{ } \checkmark\]This value MUST equal infinity given that c is not negative infinity, which it can't be given the restriction.

Answer 10

Our teacher wanted us to use the definition of a positive infinite limit tho

Answer 11

I don't understand what you mean by "the definition of a positive limit"?