lim_(x rarr0)((sin(x))/(x))

Identify Limit Function: We are asked to find the limit of the function (sin(x))/x as x approaches 0. This is a well-known limit in calculus, often referred to as the ＂sinc function＂ limit.Apply L'Hôpital's Rule: To solve this limit, we can use L'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value c results in an indeterminate form 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator and then finding the limit of the new function. However, in this case, we recognize that this limit is a standard result in calculus, and we can apply the known result without using L'Hôpital's Rule.Use Standard Result: The standard result for the limit of (sin(x))/x as x approaches 0 is 1. This is because the sine function and the linear function x are nearly identical for values of x close to 0, and their ratio approaches 1.Determine Final Limit: Therefore, the limit of (sin(x))/x as x approaches 0 is 1.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$\lim _{x \rightarrow 0}\left(\frac{\sin (x)}{x}\right)$

View step-by-step help

Home

Math Problems

Algebra 1

Evaluate integers raised to rational exponents

Full solution

\lim _{x \rightarrow 0}\left(\frac{\sin (x)}{x}\right)

Identify Limit Function: We are asked to find the limit of the function $(\sin(x))/x$ as $x$ approaches $0$ . This is a well-known limit in calculus, often referred to as the ＂sinc function＂ limit.
Apply L'Hôpital's Rule: To solve this limit, we can use L'Hôpital's Rule, which states that if the limit of $f(x)/g(x)$ as $x$ approaches a value $c$ results in an indeterminate form $0/0$ or $\infty/\infty$ , then the limit can be found by taking the derivative of the numerator and the derivative of the denominator and then finding the limit of the new function. $\newline$ However, in this case, we recognize that this limit is a standard result in calculus, and we can apply the known result without using L'Hôpital's Rule.
Use Standard Result: The standard result for the limit of $(\sin(x))/x$ as $x$ approaches $0$ is $1$ . This is because the sine function and the linear function $x$ are nearly identical for values of $x$ close to $0$ , and their ratio approaches $1$ .
Determine Final Limit: Therefore, the limit of $\frac{\sin(x)}{x}$ as $x$ approaches $0$ is $1$ .