Evaluate each limit and justify your answer.
lim x...

Question

Evaluate each limit and justify your answer.

lim x→0 (x / √16x+1-1)^1/3

Accepted Answer

Welcome back, everyone. Evaluate the limit of the function F of X. Which is equal to 4 x divided by square root of 125 x plus 1 minus 1, all raised to the power of 1/3 as X approaches 0, and were given 4 answer choices A says 0 B 2/5. C 5 halves and D does not exist. So let's write down the limit. We're constraining limit as x approaches 0. Let's write down the function 4 x divided by square root of 125 x plus 1 minus 1 all raised to the power of 1/3. If we use direct substitution, well, this gives us 0 divided by 0, which is an indeterminate form. So to avoid this, we have to notice that there is a radical term, and what we're going to do is simply multiply both sides of the fraction by the conjugate. Of the denominator. So we're going to get 4 X multiplied by square root of 125 X plus 1 + 1, right, because this is the conjugate we're replacing -1 with + 1. And now for the denominator we're going to get square root of 125 X plus 1. -1 multiplied by the conjugate, we want to make sure that our fraction is unchanged, so we also have to multiply the denominator by the conjugate. And now what we're going to do is simply finish the function. We're going to add the exponent, and we're going to apply the formula for the difference of squares in reverse, right? So we get limit as X approaches 0. Our numerator doesn't change. We get 4 X multiplied by square root of 125 X plus 1 + 1. For the denominator, we have to square the first term, which gives us 125 x plus 1 and subtract the square of the second term, which is 1 squared, and that's 1. And now 1 minus 1 gives us 0 so we can. Cancel out these. And we can notice that access will also be canceled out. Which essentially means that we may now apply the right substitution. And see if we get a finite value. So we get 4 multiplied by square root of 125 multiplied by 0 gives us 0, so square root of 1. Plus one Divided by 125. So 1 + 1 gives us 22 multiplied by 4 is 8. And then in the denominator we have 125, right? Also, let's not forget that we are raising the results the power of 1/3. So we can distribute the exponent cubic root of 8 is 2. And cubic root of 125 is 5. So the final answer is 2/5s, which corresponds to the answer choice B. Thank you for watching.

Evaluate each limit and justify your answer.
lim x→0 (x / √16x+1-1)^1/3

Key Concepts

Limits

Indeterminate Forms

Cube Roots

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