{Use of Tech} Computing limits with angles in degrees Suppos...

Question

{Use of Tech} Computing limits with angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.
b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.

b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.

Accepted Answer

Welcome back, everyone. In a trigonometry class, students are asked to compute the limit. Limit as x approaches 0 of T X divided by X, where TX represents the tangent of X degrees as X approaches 0. Which of the following options correctly represents the value of this limit api divided by 180, B 0, C 1, and the infinity. So we want to valid the limit. As X approaches 0 of TFX divided by X where TFX is tangent, right? So, essentially we're taking tangent of X degrees divided by X. Now if we use direct substitution, this gives us 0 divided by 0, so we want to simplify the expression and see if we can get a finite limit. So what we're going to do is simply rewrite tangent in terms of sine and cosine. We're going to get limit as X approaches 0 of sine of x degrees divided by cosine of x degrees. That's the definition of tangent, and then we are dividing by X. Which means multiplying by 1 divided by X, right? And now we can make use of the limit as x approaches 0 of sin of x divided by X, which is 1. We're going to use this definition in our calculation. Now notice that X is given in degrees, right? And we have to recall that. Degrees are equal to divided by 180 radiance. So essentially if we multiply both sides by X, we're going to get X degrees or equal to pi divided by 180 x radiance. So let's use our limit. We're going to get limit. As x approaches 0 of sin of x degrees, right, which essentially corresponds to I X divided by 180 radiance. Divided by X, right? So essentially what we can do here. I write the X below the sign function, right? Because we want to make use of the definition and then We are multiplying that limit by another limit, which is the limit as x approaches 0 of cosine of x degrees, right, where basically we can say cosine. Of pi x divided by 180 radiance and now notice that cosine is In the denominator. So we're going to write that in the denominator and in the numerator we are simply going to add a 1. Now let's understand that the second limit is equal to 1 because if we take X equals 0, cosine of 0 is 11 divided by 1 gives us 1. So all that we have to do is evaluate. The first limit, right? So let's go ahead and do that and let's focus on limit SX approaches 0 of sign of pi x divided by 180 divided by X. And if we want to use the definition, well, we're going to adjust the denominator to correspond to the given angle. So we're going to take pi X divided by 180. So now if we perform this addition of pi in 180, this means that to keep the original function unchanged, we have to multiply the whole function by Why Because we're going to have pi in the denominator, right, so we can cancel pi out, and then we have to divide by 180, and that's because if we have a double division, 180 goes on top, so 180 can also be canceled out, right? So this is exactly what we want to evaluate. And now we can factor out the constant. So now we have a constant which is pi divided by 180, and then we're evaluating the limit as x approaches 0 of sine of pi x divided by 180. Divided by The same Angle, right, so i X divided by 180. Which is 1. Now, why is it 1? Well, if x approaches 0, then pi x divided by 180 also approaches 0. So we fit the definition and we can say that this limit is equal to 1, meaning the final answer is our constant. So that's pi divided by 180, which corresponds to the answer choice a. Thank you for watching.

Key Concepts

Limits

Sine Function

L'Hôpital's Rule

Watch next