Finding Limit Using LHpitals Rule

Find the limit as $x \to 0$ of $\displaystyle \frac{\sin x - x}{x^2}$ using L'Hôpital's Rule.

In this problem, we are tasked with finding the limit of a function involving trigonometric terms as the variable approaches zero. The function given is the difference between a sine function and a linear term divided by a quadratic term. This type of limit often results in an indeterminate form such as 0/0, which allows us to utilize L'Hôpital's Rule.

L'Hôpital's Rule is a powerful tool in calculus that applies to situations where direct substitution in a limit results in an indeterminate form like 0/0 or infinity/infinity. When these indeterminate forms appear, the rule suggests taking the derivative of the numerator and denominator separately, and then evaluating the limit again. Applying this rule can simplify complex fractions, especially those involving periodic or oscillatory functions, as seen with trigonometric expressions.

The problem also provides an opportunity to explore the behavior of the sine function near zero and how it approximates a linear expression due to its Taylor series expansion. When employing L'Hôpital's Rule, remember that it can sometimes be necessary to apply the rule more than once if the resulting expression still yields an indeterminate form. This problem is a great example of how calculus can simplify the process of finding limits and uncovering the subtle behaviors of functions as they approach specific points, especially those that aren't immediately intuitive.

Related Problems