The limit of the function as θ approaches 0 can be found by using L'Hopital's Rule, which states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.

The function is given as (8θ + tanθ) / sinθ.

First, we need to find the derivatives of the numerator and the denominator.

The derivative of 8θ is 8, the derivative of tanθ is sec²θ, and the derivative of sinθ is cosθ.

So, the derivative of the numerator is 8 + sec²θ and the derivative of the denominator is cosθ.

Now, we substitute these into the function and take the limit as θ approaches 0.

lim θ→0 (8 + sec²θ) / cosθ

As θ approaches 0, sec²θ approaches 1 (since sec 0 = 1) and cosθ also approaches 1 (since cos 0 = 1).

So, the limit of the function as θ approaches 0 is (8 + 1) / 1 = 9.

Question

The limit of the function as θ approaches 0 can be found by using L'Hopital's Rule, which states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.

The function is given as (8θ + tanθ) / sinθ.

First, we need to find the derivatives of the numerator and the denominator.

The derivative of 8θ is 8, the derivative of tanθ is sec²θ, and the derivative of sinθ is cosθ.

So, the derivative of the numerator is 8 + sec²θ and the derivative of the denominator is cosθ.

Now, we substitute these into the function and take the limit as θ approaches 0.

lim θ→0​ (8 + sec²θ) / cosθ

As θ approaches 0, sec²θ approaches 1 (since sec 0 = 1) and cosθ also approaches 1 (since cos 0 = 1).

So, the limit of the function as θ approaches 0 is (8 + 1) / 1 = 9.

Knowee AI · Accepted Answer