Question Video: Use L’Hôpital’s Rule to Evaluate a Limit Mathematics

Video Transcript

For the functions 𝑓 of 𝑥 equals two 𝑥 squared and 𝑔 of 𝑥 equals four 𝑒 to the two 𝑥 power, evaluate the limit as 𝑥 approaches ∞ of 𝑓 of 𝑥 over 𝑔 of 𝑥 using L’Hôpital’s rule.

Let’s firstly remind ourselves of L’Hôpital’s rule. Suppose that 𝑓 and 𝑔 are differentiable and 𝑔 prime of 𝑥 is not equal to zero near 𝑎, but 𝑔 prime of 𝑎 equals zero is okay. If the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is zero and the limit as 𝑥 approaches 𝑎 of 𝑔 of 𝑥 is zero or the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is positive or negative ∞ and the limit as 𝑥 approaches 𝑎 of 𝑔 of 𝑥 is positive or negative ∞. Then the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 over 𝑔 of 𝑥 is equal to the limit as 𝑥 approaches 𝑎 of 𝑓 prime of 𝑥 over 𝑔 prime of 𝑥, as long as this limit exists or is positive or negative ∞.

Although this looks quite complicated, all it’s saying is that as long as we meet these conditions, if we evaluate a limit by direct substitution and get an indeterminate form, then the limit of the quotient of these functions is the same as the limit as the quotient of the derivatives of these functions. Now, for our question, we’ve already been told what 𝑓 of 𝑥 and 𝑔 of 𝑥 is equal to. Before we use L’Hôpital’s rule, we must check that we meet the required conditions.

Firstly, we must have that 𝑓 and 𝑔 are both differentiable. Well, we can differentiate 𝑓 of 𝑥 equals two 𝑥 squared using the power rule. That gives us that 𝑓 prime of 𝑥 equals four 𝑥. And we can differentiate 𝑔 of 𝑥 equals four 𝑒 to the two 𝑥 power using differentiation rules for exponential functions. The chain rule tells us that the derivative with respect to 𝑥 of 𝑒 raised to a function of 𝑥 power is equal to 𝑓 prime of 𝑥 multiplied by 𝑒 raised to the power of 𝑓 of 𝑥. So 𝑔 prime of 𝑥 is equal to eight 𝑒 to the two 𝑥 power. So because we managed to differentiate 𝑓 and g, we’ve satisfied that first condition.

We also need to make sure that 𝑔 prime of 𝑥 is not equal to zero near 𝑎. 𝑎 is the limit, which we we’re given in the question as ∞. So if we evaluate 𝑔 prime of 𝑥 at ∞, we can see that this is not going to be zero. So we’ve satisfied that condition too. Now, let’s double-check that we get an indeterminate form when we try to evaluate this limit directly.

The limit as 𝑥 approaches ∞ of two 𝑥 squared is just going to be ∞ and the limit as 𝑥 approaches ∞ of four 𝑒 to the two 𝑥 power is also going to be ∞. So we find ourselves in this situation. What L’Hôpital’s rule tells us is that the limit as 𝑥 approaches ∞ of two 𝑥 squared over four 𝑒 to the two 𝑥 power is the same as the limit as 𝑥 approaches ∞ of four 𝑥 over eight 𝑒 to the two 𝑥 power. But this limit just gives us ∞ over ∞, which is another indeterminate form. So we apply L’Hôpital’s rule again, but this time to this function. So by L’Hôpital’s rule, the limit as 𝑥 approaches ∞ of four 𝑥 over eight 𝑒 to the two 𝑥 power is the same as the limit as 𝑥 approaches ∞ of four over 16𝑒 to the two 𝑥 power. This is because four 𝑥 differentiated to give us four and eight 𝑒 to the two 𝑥 power differentiated to give us 16𝑒 to the two 𝑥 power.

We know that we’re okay to use L’Hôpital’s rule because both the function on the numerator and denominator are differentiable and 16𝑒 to the two 𝑥 power is not equal to zero near ∞. And we’re in this scenario where the limit as 𝑥 approaches ∞ of the function on the numerator and denominator both were ∞. So now, applying the limit as 𝑥 approaches ∞, we find that the denominator approaches ∞. So the limit of the function is zero. And that gives us our final answer.

So by applying L’Hôpital’s rule twice to our function, we were able to evaluate this limit. Remember, when solving this type of question, it’s important to check that you meet the necessary conditions to use L’Hôpital’s rule.