Use properties of limits and the following limits limx → 0(sin x)/(x)=1, limx → 0(cos x-

Name: Problem 83, Chapter 11: Precalculus
Uploaded: 2021-09-05T17:31:39-08:00
Duration: 2 min 51 s
Channel: Farnood Ensan
Description: Use properties of limits and the following limits limx → 0(sin x)/(x)=1, limx → 0(cos x-1)/(x)=0 limx → 0sin x=0, limx → 0cos x=1 to find the indicated limit. limx →0 (2 sinx+cosx-1)/(3 x)

step 1 In this question, we are given the following four limits. First, the fact that the limit as x go

Question

Use properties of limits and the following limits $$\begin{array}{lc}\lim _{x \rightarrow 0} \frac{\sin x}{x}=1, & \lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0 \\\lim _{x \rightarrow 0} \sin x=0, & \lim _{x \rightarrow 0} \cos x=1\end{array}$$ to find the indicated limit. $$\lim _{x \rightarrow 0} \frac{2 \sin x+\cos x-1}{3 x}$$

   Use properties of limits and the following limits
$$\begin{array}{lc}\lim _{x \rightarrow 0} \frac{\sin x}{x}=1, & \lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0 \\\lim _{x \rightarrow 0} \sin x=0, & \lim _{x \rightarrow 0} \cos x=1\end{array}$$
to find the indicated limit.
$$\lim _{x \rightarrow 0} \frac{2 \sin x+\cos x-1}{3 x}$$

Precalculus

Robert Blitzer 6th Edition

Chapter 11, Problem 83

Instant Answer

Solved by Expert Farnood Ensan

09/05/2021

Step 1

Step 1: We are given the limit $$\lim _{x \rightarrow 0} \frac{2 \sin x+\cos x-1}{3 x}$$ We can split this limit into two separate limits: $$\lim _{x \rightarrow 0} \frac{2 \sin x}{3x} + \lim _{x \rightarrow 0} \frac{\cos x-1}{3x}$$ Show more…

Show all steps

Thanks for your feedback!

Use properties of limits and the following limits $$\begin{array}{lc}\lim _{x \rightarrow 0} \frac{\sin x}{x}=1, & \lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0 \\\lim _{x \rightarrow 0} \sin x=0, & \lim _{x \rightarrow 0} \cos x=1\end{array}$$ to find the indicated limit. $$\lim _{x \rightarrow 0} \frac{2 \sin x+\cos x-1}{3 x}$$

Play audio

Feedback

Farnood Ensan and 101 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Properties of Limits

This concept refers to the ability to split the limit of a sum or difference of functions into the sum or difference of their limits, and to factor out constants from limits. This property is crucial when dealing with complex limit expressions, as it allows one to break down an expression into simpler parts whose limits are easier to compute.

Standard Trigonometric Limits

In solving limits involving sine and cosine functions, certain standard limits are frequently used. Two well-known limits are lim??0 (sin x)/x = 1 and lim??0 (cos x ? 1)/x = 0. These limits are key to evaluating expressions that arise in calculus, especially when small-angle approximations or perturbative approaches are applicable.

Please give Ace some feedback