Use an appropriate limit definition to prove the following lim...

Question

Use an appropriate limit definition to prove the following limits.

lim x→ 5x^2 − 25 / x − 5=10

Accepted Answer

Welcome back, everyone. Select the correct relationship between epsilon and Delta to prove limit as X approaches 8 of X2 minus 64 divided by X minus 8 is equal to 16, using the epsilon-delta definition of a limit. We are given 4 answer choices. A says delta equals epsilon divided by 8. B delta equals epsilon divided by 3. C equals 2 epsilon, and D delta equals epsilon. So to prove that this limit is equal to 16 using the epsilon-delta definition, we need to show that for every epsilon greater than 0. There exists. Delta that is also greater than 0 such that If 0 is less than the absolute value of X minus A, Less than Delta Then The absolute value of F of X minus L, where L is the value of the limit. Must be less than Epsil. So now let's define all of the variables. A is equal to 8 because X approaches 8. So we have X approaches 8. This gives us our A value. Now L is the value of the limit, which is 16, and the given function F of X is X2 minus 64 divided by X minus 8. So what we're going to do is analyze the second inequality, and we're going to substitute F of X, that's X2 minus 64 divided by X minus 8, and we need to subtract the value of the limit, which is 16. This must be less than epsilon. From here we can simplify and let's start simplifying by performing factorization of the difference of squares. That's X2 minus 82, which is X minus 8 multiplied by x + 8. And this is divided by X minus 8. And then we're subtracting 16. We can notice that we can cancel out the common factor. And simplify that to X + 8 minus 16. Which gives us the absolute value of X minus 8, and this must be less than epsilon. Now if we look at the first inequality, we Will have 0 less than the absolute value of x minus 8. Which must be less than Delta. And now let's compare. Well, we have an inequality which says that the absolute value of x minus 8 is less than delta, and also the absolute value of X minus 8 must be less than delta. So less than epsilon, less than Delta. This means that if we have the same quantity on the left hand side of each inequality. The quantities on the right hand sides of each inequality will be equal, so we can choose delta. Equals epsilon. And therefore, the correct answer to this problem corresponds to the answer choice D. Thank you for watching.

Use an appropriate limit definition to prove the following limits.

lim x→ 5x^2 − 25 / x − 5=10

Key Concepts

Limit Definition

Factoring Polynomials

Indeterminate Forms

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