Finding Limits
In Exercises 9–24, find ...

Question

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.

lim h →0 ((x + h)² ― x²)/h

Accepted Answer

Welcome back, everyone. Determine the limit. Limit as H approach is 0 of X + 3 H2 minus X2 divided by 2H. We're given 4 answer choices A says 6 x + 9, B 6 X C 3 X + 9, and D 3 X. So let's write down the given limit. Lim as H approaches 0. In the numerator we have X + 3 H. This term is squared. We are subtracting X2, and we are dividing by 2 H. If we simply attempt direct substitution, we're going to get X plus 3 H2. So since H approaches 0, we simply get X2 minus X2. This is 0, and in the denominator 2 multiplied by 0 gives us 0 as well. And this is an indeterminate form, so we cannot really evaluate it yet, but we can factor out the numerator because it corresponds to the difference of squares. So what we're going to do is simply take our first term X + 3 H. Subtract the second term, which is X. Let's ignore the squares, right, because we're performing factorization a2 minus B2, which is equal to a minus B multiplied by A plus B. Our A is X +3 H and our B is X. So we get X + 3H minus X multiplied by X + 3 H + X, and all of that is divided by 2 H. Now let's notice that in the first factor we can cancel out X minus X. And now we can just simplify. This gives us a limit. As H approaches 0 of 3 H multiplied by 2 X plus 3 H. Divided by 2 H and now we have canceled out the whole right because we have h as a factor in the numerator and the denominator. So once again from here we can attempt direct substitution we get limit as H approaches 0. Of what we can do, we can simply factor out three halves in front of the limit, and we are left with 2 X plus 3 H. This gives us 3 halves in 2 X plus. We multiplied by 0, right, because each approach is 0. So overall, 3 multiplied by 0 is 0. And we get 2 Xs multiplied by 3 divided by 2. So 2 and 2, those can be canceled out. And we simply get we act. Which corresponds to the answer choice the. Thank you for watching.

Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.

lim h →0 ((x + h)² ― x²)/h

Key Concepts

Limits

Difference Quotient

Derivative

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