Find the limit, if it exists. (If an answer does not exist, enter DNE.)
lim x→∞ (sqrt(49x^2+x)−7x)
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.
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!
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
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).