17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞...

Question

17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.

Accepted Answer

Welcome back everyone. Determine whether the following statement is true or false. As x approaches infinity, F of X equals square root of 5 X + 3 grows faster than G of X equals square root of X + 3 A as true, and Bs as false. For this problem, we want to have a reference point, so we want to assume that this statement is true, right? So, if this statement is true, then we can define limit as X approaches infinity. Of G of X divided by F of X. Equal to 0, because if this statement is true, then F of X grows faster, meaning denominator dominates over the numerator and the limit is going to be zero. We want to check if this statement is true, right? So if we get a limit of 0, we can verify that the statement is true. If it's not zero, the statement is false. So, let's go ahead and substitute each function into the limit. We're going to get limit as X approaches infinity. Of square root of x + 3 divided by square root of 5 X + 3. Each function approaches infinity, as X approaches infinity, so we basically get an indeterminate form. What we can do is apply the properties of radicals. We can just rewrite it as square root of our ratio X + 3 divided by 5 x + 3, and I will consider this rational expression under the radical. What we're going to do is simply factor out X. For the numerator and the denominator. So we are going to get limit as X approaches infinity. Of square root of x in 1 + 3 divided by X. Divided by X in par 5 + 3 divided by X. We can cancel out each X, and as X approaches infinity, 3 divided by X is going to approach 0 because we're dividing a constant by an infinitely large number, so the limit is 0, and we're basically left with square root of 15th, right? We get square root of 1 divided by square root of 5, which is basically 1 divided by square root of 5, or square root of 5 divided by 5 if we want to rationalize the denominator. This is not equal to 0. We got a constant, which essentially means that these functions grow at the same rate. Remember that if we get a constant that is not equal to 0, right? The two functions grow at the same rate. So the answer to this problem is be false. Thank you for watching.

17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.

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