17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (eˣ - x - 1) / 5x²

Accepted Answer

Welcome back, everyone. Apply Lapto's rule to find the limit of the following function as X approaches 0. F of X is equal to 2 to the power of X minus 2X minus 2 divided by 3x quad. Were given 4 answer choices A2, B 2/3, C1, and the 13. So let's go ahead and write down the limit. Limit as x approaches 0 of F of X, which is 2 e's the power of X minus 2X minus 2. The all divided by a 3 x squad. As always, we begin with the direct substitution. We're going to get 2 E to the power of 0, minus 2 multiplied by 0 minus 2, all divided by 3 multiplied by 0 squared. In the numerator, we are going to get 2 minus 0 minus 2. This gives us 0, and in the denominator, we have 0 as well. For this indeterminant form, we can apply Lapito's rule because we have 0 divided by 0. And the rule says that we can differentiate the numerator of the fractions. We're going to take the derivative of 2 E to the power of X. -2 X minus 2. And divide that By the derivative of 3 x 2. So we have applied Lapto's rule, and we're saying that since we have this in the terminal form, we want to identify the derivative of the numerator and the denominator. And then attempt the right substitution once again. So the derivative of EX is EX. This gives us 2 E to the power of X, the derivative of negative to X is -2, and the derivative of -2 is 0. And we are taking the derivative of 3 X2, which is 6 X. Now let's attempt the direct substitution once again. We're going to get 2 E to the power of 0 minus 2 divided by 6 multiplied by 0. Once again, this is 0 divided by 0, right? Let's state that we're getting 0 divided by 0, which means that once again we want to apply Lapital's rule, and we're going to perform differentiation once again. Starting with the numerator to eats the power of x minus 2, we want to differentiate that, and for the denominator we are differentiating 6 X. This gives us limit as x approaches 0 of 2 to the power of X. The derivative of -20 and the derivative of X is 1, so we only have 6. In the denominator Now let's attempt the right substitution once again. So this gives us 2 Es, the power of 0, divided by 6, which is 2 multiplied by 1 divided by 6 or 2 divided by 6. This is already a finite number. We want to simplify it, so we are going to get 13 when we divide both sides by 2. Now, since this is not an indeterminate form, we can conclude that the correct answer to the problem corresponds to the answer choice D. 1/3 is the limit. Thank you for watching.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (eˣ - x - 1) / 5x²

Key Concepts

Limits

L'Hôpital's Rule

Exponential Functions

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