How do you find the limit of #(sin(x)/3x) # as x approaches 0 using l'hospital's rule?

Answer 1

#lim_(x->0)(sin(x)/(3x)) = 1/3.#

See explanation below.

#lim_(x->0)(sin(x)/(3x)) = "0/0"# so we can use Bernouilli L'Hôpital's rule.
#lim_(x->0)(sin(x)/(3x)) = lim_(x->0)((sin(x)')/((3x)')) =lim_(x->0) cos(x)/3 = 1/3.#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the limit of (sin(x)/3x) as x approaches 0 using L'Hospital's rule, we can differentiate the numerator and denominator separately.

Differentiating the numerator, we get cos(x). Differentiating the denominator, we get 3.

Now, we can evaluate the limit of the derivative of the numerator divided by the derivative of the denominator as x approaches 0.

The limit of cos(x)/3 as x approaches 0 is equal to cos(0)/3, which simplifies to 1/3.

Therefore, the limit of (sin(x)/3x) as x approaches 0 using L'Hospital's rule is 1/3.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To find the limit of ( \frac{\sin(x)}{3x} ) as ( x ) approaches 0 using L'Hôpital's Rule:

  1. Take the derivative of the numerator and the derivative of the denominator separately.
  2. Evaluate both derivatives at ( x = 0 ).
  3. Take the limit of the ratio of the derivatives as ( x ) approaches 0.

Derivative of ( \sin(x) ) with respect to ( x ) is ( \cos(x) ). Derivative of ( 3x ) with respect to ( x ) is ( 3 ).

At ( x = 0 ), ( \sin(0) = 0 ) and ( 3(0) = 0 ).

Thus, the limit becomes ( \lim_{x \to 0} \frac{\cos(x)}{3} ).

At ( x = 0 ), ( \cos(0) = 1 ).

So, the limit is ( \frac{1}{3} ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7