Question

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

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.

Answer 2

[AI-generated answer — Clown Bot]

Well, let's see here. As x approaches infinity, we can simplify the expression and get rid of the square root by dividing everything by x.

(sqrt(49x^2+x) - 7x) / x = ((sqrt(49x^2+x)) / x) - 7

Now, let's focus on the term (sqrt(49x^2+x)) / x. As x approaches infinity, the x term becomes insignificant compared to the 49x^2 term. So we can disregard the x in the denominator. Now what we have left is:

(sqrt(49x^2+x)) / x ≈ sqrt(49x^2) / x = 7

So the limit as x approaches infinity is 7!

That wasn't too complex, was it? Just like trying to explain a joke - sometimes it's best to keep it simple!

Answer 3

[AI-generated answer — Step-by-Step Bot]

To find the limit of the given function, we can simplify it by applying some algebraic techniques. Let's begin by simplifying the expression inside the square root:

sqrt(49x^2 + x) - 7x

= sqrt(49x^2 + x) - sqrt(49x^2)

= sqrt(49x^2 + x) - sqrt(7^2 * x^2)

= sqrt(49x^2 + x) - sqrt(7^2) * sqrt(x^2)

= sqrt(49x^2 + x) - 7 * |x|

Now, we can rewrite the original limit using the simplified expression:

lim x→∞ (sqrt(49x^2+x)−7x) = lim x→∞ (sqrt(49x^2 + x) - 7 * |x|)

Next, we can deal with the two terms separately:

1. For the term √(49x^2 + x), as x approaches infinity, the x term becomes insignificant compared to x^2. So we can approximate it as √(49x^2) = 7x.

2. For the term 7 * |x|, as x approaches infinity, the absolute value sign becomes unnecessary since x is positive. So we can simply write it as 7x.

Therefore, the simplified limit expression becomes:

lim x→∞ (sqrt(49x^2+x)−7x) = lim x→∞ (7x - 7x)

Simplifying further:

lim x→∞ (7x - 7x) = lim x→∞ (0)

The limit of the function is 0. So,

lim x→∞ (sqrt(49x^2+x)−7x) = 0

Answer 4

[AI-generated answer — Explain Bot]

To find the limit as x approaches infinity, we can use algebraic manipulation and apply some limit properties.

Let's start by simplifying the expression sqrt(49x^2 + x) - 7x:

lim x→∞ (sqrt(49x^2+x)−7x)

As x approaches infinity, the term x becomes negligible compared to x^2. Therefore, we can ignore it in the expression sqrt(49x^2 + x).

lim x→∞ (sqrt(49x^2+x)−7x)
= lim x→∞ (sqrt(49x^2+x)) - lim x→∞ (7x)

Next, we simplify each term separately:

lim x→∞ (sqrt(49x^2+x)) - lim x→∞ (7x)
= lim x→∞ (sqrt(49x^2+x)) - lim x→∞ (7) * lim x→∞ (x)

Since lim x→∞ (7) = 7 and lim x→∞ (x) = ∞, we have:

lim x→∞ (sqrt(49x^2+x)) - lim x→∞ (7x)
= lim x→∞ (sqrt(49x^2+x)) - 7 * ∞

Now, we check whether lim x→∞ (sqrt(49x^2+x)) exists.

Notice that as x approaches infinity, the term 49x^2 becomes the dominant term in sqrt(49x^2 + x). Therefore, we can focus on the square root of 49x^2.

lim x→∞ (sqrt(49x^2+x)) - 7 * ∞
= lim x→∞ (sqrt(49x^2))

Since sqrt(49x^2) = 7x, we have:

lim x→∞ (sqrt(49x^2+x)) - 7 * ∞
= lim x→∞ (7x) - 7 * ∞
= 7 * ∞ - 7 * ∞

The expression 7 * ∞ cannot be determined because it is an indeterminate form. In this case, the limit does not exist (DNE), as we cannot obtain a definite value for the expression.

Therefore, the limit of (sqrt(49x^2 + x) - 7x) as x approaches infinity does not exist (DNE).