# Finding the limit of the sum of two terms that individually have different limits

For instance:

1/(e^.5t) + 1/(e^-7t)

As t grows larger, the left term goes to 0, but the right term goes to infinity.

Would I be correct in saying that the limit of the sum is infinity because the (absolute value of the) coefficient of t in the term that tends to infinity is larger than the coefficient of t in the term that tends to 0?

What if both t's had the same coefficient?

HallsofIvy
Homework Helper
It doesn't matter what the coefficients are. If $\{a_n\}$ and $\{b_n\}$ are any two sequences such that $\lim_{n\to\infty} a_n= \infty$ and $\lim_{n\to\infty}b_n= 0$, then $\lim_{n\to\infty} (a_n+ b_n)= \infty$.

More generally, if $\lim_{n\to\infty} a_n= A$ and $\lim_{n\to\infty}b_n= 0$, then $\lim_{n\to\infty} (a_n+ b_n)= A$.

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Mark44
Mentor
Fixed you LaTeX script. There were [ itex] tags mixed in with [ math] tags.
It doesn't matter what the coefficients are. If $\{a_n\}$ and $\{b_n\}$ are any two sequences such that $\lim_{n\to\infty} a_n= \infty$ and $\lim_{n\to\infty}b_n= 0$, then $\lim_{n\to\infty} (a_n+ b_n)= \infty$.

More generally, if $\lim_{n\to\infty} a_n= A$ and $\lim_{n\to\infty}b_n= 0$, then [tex]\lim_{n\to\infty} (a_n+ b_n)= A[/itex].

HallsofIvy