Limits of Average Rates of Change

Question

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = 1/x, x = -2

Accepted Answer

Welcome back, everyone. Determine the limit of the expression F of X plus H minus F of X divided by HS H approaches 0 for the function F of X equals 4 divided by 3 X and evaluated at X equals 4. We're given 4 answer choices A says -4/3, B 1/3, C-112, and D-148. So first of all, let's write down the limit. Limit is H approaches 0 of F of X plus H minus F of X divided by H. Now, let's consider each term. We're going to rewrite the limit as H approaches 0 of F of X plus H, which means that we are taking our function and replacing X with X + H. Then we are subtracting the function, which is 4 divided by 3 X, and then we are dividing by H. So now we want to combine the two fractions into one, which gives us limit as H approaches 0. Let's write the fraction sign and let's find the common denominator. Now, the common denominator is going to be 3 X multiplied by X plus H. So the first fraction needs to be multiplied and divided by X, which gives us 4 X. Divided by 3 X multiplied by X plus H, and then we are subtracting 4. Which needs to be multiplied by X plus H, because we already have our common denominator of 3 X multiplied by X plus H, right? So the second fraction is multiplied and divided by X plus H, and then we are dividing by H. In our denominator. This gives us limit as each approach is 0. Now what we're going to do is simply understand that we have a double division, so we can combine the two denominators into one, which gives us 3 XH multiplied by X plus H in the denominator, and for the numerator, let's distribute. This gives us 4 x minus 4 x minus 4 H. Now we can see that we can cancel out 4 x minus 4X, this gives us 0, right? And then our limit becomes. Limit as H approaches 0 of. Negative 4 H. Divided by 3 XH multiplied by X plus H. From here we can cancel out DH as a common factor. This gives us limit as H approaches 0 of -4 divided by 3x multiplied by x plus H. And now because each approaches 0, we're going to perform direct substitution which gives us -4. Divided by 3 X multiplied by X plus 0, or simply X, right? So 3 X multiplied by X gives us 3 X squared. So we have evaluated the limit. It's -4 divided by 3 X2, and now all that we have to do is simply evaluate -4 divided by 3 X 2 at X equals 4. This gives us -4 divided by 3 multiplied by 4 squad, which is equal to -4. Divided by 3 multiplied by 16, which is 48. And now dividing both sides by 4, we end up with -1 divided by 12. So the answer is -112, which corresponds to the answer choice C. Thank you for watching.

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = 1/x, x = -2

Key Concepts

Limits

Derivatives

Secant and Tangent Lines

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