Problem 26 Evaluate each limit (if it exist... [FREE SOLUTION]

Chapter 27: Problem 26

Evaluate each limit (if it exists). Use L'Hospital's rule (if appropriate). $$\lim _{x \rightarrow \pi / 2} \frac{2 \cos x}{x-\frac{\pi}{2}}$$

Short Answer

Expert verified

The limit is -2.

Step by step solution

Identify the indeterminate form

First, substitute $ x = \frac{\pi}{2} $ into the expression to see if an indeterminate form exists. Substituting, we get:\[\frac{2 \cos \left( \frac{\pi}{2} \right)}{\frac{\pi}{2} - \frac{\pi}{2}} = \frac{2 \cdot 0}{0} = \frac{0}{0}\]This is an indeterminate form of type $ \frac{0}{0} $, so L'Hospital's rule can be applied.

Differentiate the numerator and the denominator

Apply L'Hospital's rule, which states that if the limit is of the form $ \frac{0}{0} $, then:\[\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\]Differentiate the numerator $2 \cos x$ to get $-2 \sin x$, and the denominator $x-\frac{\pi}{2}$ to get $1$.

Evaluate the new limit

The new limit is:\[\lim_{x \to \frac{\pi}{2}} \frac{-2 \sin x}{1}\]Substitute $ x = \frac{\pi}{2} $ into the expression:\[-2 \sin \left( \frac{\pi}{2} \right) = -2 \cdot 1 = -2\]Therefore, the limit is $-2$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Limits

Limits are fundamental in calculus, helping us understand how a function behaves as its input approaches a particular value. Essentially, we examine what happens to function values as the variable `x` gets closer to some number, say `c`. For example, if we have a function $f(x) = \frac{2\cos x}{x-\frac{\pi}{2}}$ and we want to explore its behavior as $x$ nears $\frac{\pi}{2}$, we are evaluating a limit.

When approaching a limit, the exact value of the function at $x = c$ is often not important.
We are more interested in the behavior of the function as it gets infinitely close to $c$.
Limits allow us to make sense of situations where direct substitution could result in undefined expressions such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

Understanding limits sets the foundation for concepts like continuity, derivatives, and integrals, which are essential tools in calculus for analyzing and interpreting real-world phenomena.

Dealing with Indeterminate Forms

Indeterminate forms arise during limit evaluation when substitution of the limit value results in an expression like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. These forms don't provide clear information about the limit by themselves.

There are several types of indeterminate forms, including $\frac{0}{0}$, $\infty - \infty$, $0 \times \infty$, $1^\infty$, $\infty^0$, $0^0$, and $\frac{\infty}{\infty}$.
In the given exercise, substituting $x = \frac{\pi}{2}$ into the function $\frac{2\cos x}{x-\frac{\pi}{2}}$ resulted in the $\frac{0}{0}$ form.
This indicates that to evaluate the limit, we need a different approach since direct substitution was not successful.

Finding the indeterminate form signals the need for techniques like L'Hospital's rule to resolve and accurately evaluate the limit.

Applying L'Hospital's Rule

L'Hospital's rule is a powerful technique in calculus for evaluating limits that result in indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. This rule provides a method to simplify and solve these limits by differentiating the numerator and the denominator separately.

The original problem: $\lim_{x \to \frac{\pi}{2}} \frac{2 \cos x}{x-\frac{\pi}{2}}$, exhibits the $\frac{0}{0}$ form when substituting $x = \frac{\pi}{2}$.
According to L'Hospital's rule, we differentiate the numerator to get $-2 \sin x$ and the denominator $x-\frac{\pi}{2}$ turns into $1$.
The limit then simplifies to $\lim_{x \to \frac{\pi}{2}} \frac{-2 \sin x}{1}$, which is much simpler to evaluate.
Substituting $x = \frac{\pi}{2}$ results in $-2 \times 1 = -2$.

The result, $-2$, confirms the utility of L'Hospital's rule in transforming complex limit evaluations into more straightforward calculations. This technique is particularly useful in advanced calculus, ensuring limits of indeterminate forms can be solved methodically.

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91影视

Evaluate each limit (if it exists). Use L'Hospital's rule (if appropriate). $$\lim _{x \rightarrow \pi / 2} \frac{2 \cos x}{x-\frac{\pi}{2}}$$

Short Answer

Step by step solution

Identify the indeterminate form

Differentiate the numerator and the denominator

Evaluate the new limit

Key Concepts

Understanding Limits

Dealing with Indeterminate Forms

Applying L'Hospital's Rule

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