Question 4 20 pts when evaluating the limit of a function at...

Question 4 20 pts When evaluating the limit of a function at infinity, which of the following statements is true? If the degrees of the numerator and denominator are the same, the limit is the ratio of the leading coefficients. All of the above. If the degree of the denominator is greater than the degree of the numerator, the limit is zero. If the degree of the numerator is greater than the degree of the denominator, the limit is infinity or negative infinity. Question 4 20 pts K When evaluating the limit of a function at infinity,which of the following statements is true? If the degrees of the numerator and denominator are the same,the limit is the ratio of the leading coefficients Q All of the above O If the degree of the denominator is greater than the degree of the numerator, the limit is zero. O If the degree of the numerator is greater than the degree of the denominator, the limit is infinity or negative infinity

Transcript

00:01 Okay, we have a couple of limit questions here, and i have a feeling these are going to be about lopi tals rule for a reason that i will show you shortly.

00:14 So our first one is limit as theta approaches zero of the function tangent of 7 theta divided by sine of 4 theta.

00:42 So the reason i think we're going to have to use lopitals rule here is because if you substitute zero for theta, you're going to get tangent of zero, which is zero, and you're going to get sign of zero, which is zero.

01:01 So you're going to get zero divided by zero.

01:03 And that is not okay.

01:04 We don't like that.

01:06 But that means we can use lopetal's rule.

01:08 So let's take the derivative of the numerator and the denominator.

01:12 So the derivative of tangent of 7 theta, derivative of tangent, i believe is 1 over cosine squared, or we can just write that as secant squared of theta.

01:33 So we can write this as derivative of tangent is secant squared of 7 theta times 7, right, because of the chain rule.

01:47 And we're going to divide that by the derivative of sine, which i believe is just going to be cosine.

02:08 So this is going to be cosine of four theta times.

02:17 Okay.

02:18 And now let's try to evaluate theta at zero.

02:23 So in the numerator there, we are going to get secant squared of zero.

02:35 Um, secant of zero is one.

02:40 Um, so that is just going to be one squared, which is one times seven, which is seven.

02:48 Um, similarly, uh, cosine of zero is also one.

02:58 So we're just going to get one times four, which is four.

03:02 Um, and the limit here, let me try that one more time.

03:08 Uh, the limit as theta approaches zero of this first function is just going to be seven fourths.

03:14 Basically, if anything involves sign, it's going to be zero.

03:18 If anything involves cosine and you're evaluating at zero, it's going to be one.

03:23 That's something to remember about these.

03:28 And now let's look at the second one.

03:29 Second one's going to be very similar.

03:34 Okay, second one is limit as theta approaches zero of sine squared of three theta divided by theta squared.

04:05 Similarly for this one, if you just go ahead and substitute zero right away, you're going to get zero over zero.

04:11 So we're going to have to use lopi -tal's rule again.

04:14 If we take the derivative of this first one, you are going to get two.

04:18 You're going to have to use the power rule first.

04:22 Two sine of three theta.

04:25 The derivative of sine of three theta is going to be cosine of three.

04:31 Data and then you have to multiply by three...

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