Find the limit, if it exists. (If an answer does not exist, enter DNE.)

Question

lim x→∞ (sqrt(49x^2+x)−7x)

Answer 1

There is a pretty standard way of doing this sort of problem.
Keep it in mind when you encounter this sort of expression.
Write
sqrt(49x^2+x)-7x=[sqrt(49x^2+x)-7x][sq...
=[49x^2+x-49x^2]/[sqrt(49x^2+x)+7x]
=x/[sqrt(49x^2+x)+7x]
=1/[sqrt(49+1/x)+7]-> 1/14 as x->+infinity.

Note here, dividing num and den of the following by x
x/[sqrt(49x^2+x)+7x]=1/[sqrt{49x^2+7x)...
=1/[sqrt(49+1/x)+7].

So the limit is 1/14.
Hope that helps.