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

Question

Evaluate each limit and justify your answer.

lim t→4 t−4 /√t−2

Accepted Answer

Welcome back, everyone. Evaluate the limit of the function F of X is equal to x minus 49 divided by square root of X minus 7 S X approaches 49. We're given four answer choices A says 0, B7, C 14, and D does not exist. So let's ride the limit. Limits X approaches 49. In the numerator we have x minus 49. In the denominator that's square root of x minus 7. Let's begin with direct substitution, which gives us 49 minus 49 in the numerator. In the denominator, we get square root of 49 minus 7. And this is an indeterminant form because we get 0 divided by 0, right? Because this is an indeterminate form we want to. Eliminate a potential hole of this rational expression, and we can do that by using factorization. Let's notice that our numerator is X minus 49. And we can actually rewrite it as square root of X. Squared Minus 72 and this is a form that can be factored out using the formula of the difference of two squares. The factor form would be square root of X minus 7, multiplied by square root of X + 7. So based on the formula, we now have a factored form of the numerator. And now we can write the limit as X approaches 49 of square root of X minus 7 multiplied by square root of X + 7. Divided by Square root of x minus 7. And now we can notice that we have a common factor that we can cancel out, so we have successfully identified a hole in this function, and now we can apply the direct substitution once again. So we are only left with square root of X + 7. Applying the direct substitution, we get square root of 49 + 7. And now this simply gives us 7 + 7, which is 14. So the final answer to this problem corresponds to the answer choice C. Thank you for watching.

Evaluate each limit and justify your answer.
lim t→4 t−4 /√t−2

Key Concepts

Limits

Indeterminate Forms

Rational Functions

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