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Magazine Chapter 1: Problem 33
Evaluate \(\lim _{x \rightarrow 0} \frac{\sin x-x-x \cos x+x^{2} \cot x}{x^{5}}\)
Short Answer
Expert verified
The short answer is: \( \lim_{x\rightarrow 0} \frac{\sin x - x - x\cos x + x^2\cot x}{x^5} = 0 \).
Step by step solution
01
Rewrite cot(x) with cos(x) and sin(x)
We will start by rewriting the x^2 cot(x) term as x^2 times the quotient of cos(x) and sin(x):
\( \lim_{x\rightarrow 0} \frac{\sin x - x - x\cos x + x^2\frac{\cos x}{\sin x}}{x^5} \)
02
Divide terms by x
Now, we will divide every term in the numerator by x. This will help us rewrite the expression in a more manageable form:
\( \lim_{x\rightarrow 0} \frac{x(\frac{\sin x}{x} - 1 - \cos x + x\frac{\cos x}{\sin x})}{x^5} \)
03
Factor x from the denominator
Now we will factor x from the denominator:
\( \lim_{x\rightarrow 0} \frac{(\frac{\sin x}{x} - 1 - \cos x + x\frac{\cos x}{\sin x})}{x^4} \)
04
Evaluate known limits
Next, we use the known limit, \( \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1 \). We will substitute this into our expression:
\( \lim_{x\rightarrow 0} \frac{(1 - 1 - \cos x + x\frac{\cos x}{\sin x})}{x^4} \)
05
Simplify expression
The expression simplifies to:
\( \lim_{x\rightarrow 0} \frac{(- \cos x + x\frac{\cos x}{\sin x})}{x^4} \)
06
Factor cos(x) from the numerator
Now, we will factor cos(x) from the numerator:
\( \lim_{x\rightarrow 0} \frac{\cos x(-1 + \frac{x}{\sin x})}{x^4} \)
07
Separate into two limits
Now, we can separate the expression into two different limits:
\( \lim_{x\rightarrow 0} (\frac{\cos x}{x^4}) \times \lim_{x\rightarrow 0} (-1 + \frac{x}{\sin x}) \)
08
Evaluate limits
We know that as x approaches 0, the cos(x) term is 1. Therefore, the first limit is:
\( \lim_{x\rightarrow 0} \frac{\cos x}{x^4} = \frac{1}{0} = \infty \)
The second part converges to \(-1\), since we know the limit \(\lim_{x\rightarrow 0} \frac{x}{\sin x} = \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1\), and so we can substitute 1 back:
\( \lim_{x\rightarrow 0} (-1 + \frac{x}{\sin x}) = (-1 + 1) = 0 \)
09
Final answer
Now, we can combine the results:
\( \infty \times 0 = 0 \)
Therefore, the limit is:
\( \lim_{x\rightarrow 0} \frac{\sin x - x - x\cos x + x^2\cot x}{x^5} = 0 \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit of a Function
In calculus, limits help us understand the behavior of functions as the input approaches a certain value. They tell us what value the function approaches as the input gets closer and closer to a point. For the expression \( \lim_{x \rightarrow 0} \frac{\sin x - x - x \cos x + x^2 \cot x}{x^5} \), the limit signifies what the overall expression tends towards when \( x \) is approaching zero.
- This is useful when dealing with expressions that become undefined or indeterminate at certain points, such as when a function divides by zero.
- Evaluating limits forms the foundation of derivative calculations, which represent the rate of change of functions.
Trigonometric Functions
Trigonometric functions like \( \sin x \), \( \cos x \), and \( \cot x \) are vital in calculus for analyzing periodic functions. These functions model wave patterns, circles, and oscillations, which are essential in numerous practical applications.
- \( \sin x \) (sine) and \( \cos x \) (cosine) are fundamental trigonometric functions that describe the ratios of a right-angled triangle based on an angle.
- \( \cot x \) is the cotangent function, defined as the reciprocal of the tangent, or the ratio \( \cot x = \frac{\cos x}{\sin x} \).
L'H么pital's Rule
L'H么pital's Rule is a powerful tool in calculus used to solve limits that result in indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When limits are difficult to evaluate through algebraic manipulation, this rule helps by differentiating the numerator and the denominator.
To apply L'H么pital's Rule:
In problems like \( \lim_{x \rightarrow c} \frac{f(x)}{g(x)} \) where both \( f(c) \) and \( g(c) \) evaluate to zero or infinity, applying this rule might provide a clearer solution.
To apply L'H么pital's Rule:
- Differentiate the numerator and the denominator separately.
- Then, re-evaluate the limit of the new fraction.
In problems like \( \lim_{x \rightarrow c} \frac{f(x)}{g(x)} \) where both \( f(c) \) and \( g(c) \) evaluate to zero or infinity, applying this rule might provide a clearer solution.
Mathematical Proofs
Mathematical proofs are logical arguments that verify statements or solve problems by using a sequence of justified steps. Understanding proofs is important in calculus as it builds a solid foundation for knowing why a particular solution is correct.
- Proofs rely on assumptions, axioms, and previously established theorems.
- They involve a chain of reasoning from known facts to establish the truth of a statement.
