Question

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow 0} \frac{3 \sec ^{5} x}{x^{2}+4}$$

Short answer

Answer: The value of the limit is \(\frac{3}{4}\).

Step-by-step solution

Rewrite the expression in terms of cosine

Recall that \(sec(x) = \frac{1}{cos(x)}\). So, the expression can be rewritten as: $$\lim _{x \rightarrow 0} \frac{3 \left(\frac{1}{\cos x}\right)^{5}}{x^{2}+4}$$ Which simplifies to: $$\lim _{x \rightarrow 0} \frac{3}{\cos^{5} x \left(x^{2} + 4\right)}$$

Evaluate the limit

Now, we evaluate the limit as x approaches 0: $$\lim _{x \rightarrow 0} \frac{3}{\cos^{5} x \left(x^{2} + 4\right)} = \frac{3}{\cos^{5}(0) \left(0^{2} + 4\right)}$$ As \(\cos(0) = 1\), then: $$\frac{3}{\cos^{5}(0) \left(0^{2} + 4\right)} = \frac{3}{1^{5} \left(4\right)} = \frac{3}{4}$$ So, the limit exists and is equal to: $$\boxed{\lim _{x \rightarrow 0} \frac{3 \sec ^{5} x}{x^{2}+4} = \frac{3}{4}}$$

Key concepts

Limits of Functions

Limit evaluation is a foundational concept in calculus, essential for understanding how functions behave as they approach a specific point. The limit of a function as the variable approaches a certain value is the expected value that the function should get to as you get closer to that point.

For instance, when you see an expression like \(\lim_{x \rightarrow c} f(x)\), it represents the limit of the function \(f(x)\) as \(x\) approaches the value \(c\). If the function reaches a specific value \(L\) as \(x\) approaches \(c\) from both sides, we can say that \(\lim_{x \rightarrow c} f(x) = L\). However, if the function does not approach a specific value, the limit does not exist. Calculating limits is crucial for understanding the continuity of functions, the derivative, and the integral--all of which are significant concepts in calculus.

Trigonometric Limits

Trigonometric limits involve limit operations applied to trigonometric functions like sine, cosine, and tangent. These limits play a crucial role in calculus, often forming the backbone of more complex problems in differential and integral calculus.

For example, a well-known limit is \(\lim_{x \rightarrow 0}\frac{\sin x}{x} = 1\), which is a fundamental limit used in derivative calculations of trigonometric functions. In trigonometric functions, it is often useful to express the functions in terms of sine and cosine, as these are the most fundamental trigonometric functions and their limits at \(x = 0\) are well-established. Understanding these basic limits can greatly help in cracking more complicated limit problems involving trigonometric functions.

Secant Function

The secant function, denoted as \(\sec(x)\), is one of the trigonometric functions and is defined as the reciprocal of the cosine function, or \(\sec(x) = \frac{1}{\cos(x)}\). Unlike the cosine function, which has a range from -1 to 1, the secant function's range is from \(\infty\) to -1 and from 1 to \(\infty\), because as the cosine function approaches zero, the secant function tends towards infinity.

This property of the secant function is essential to understand, especially when evaluating limits involving secant. One must be cautious when dealing with limits as \(x\) approaches values where \(\cos(x)\) is equal to 0, as this would make the secant function undefined and the limit may not exist.

Calculus

Calculus is a branch of mathematics that deals with rates of change (differential calculus) and the accumulation of quantities (integral calculus). These concepts are built upon the foundation of limits, which is why mastering limits evaluation like in our example, is so important.

In differential calculus, we find the derivative of a function to determine the rate at which the function's value changes. In integral calculus, we find the area under the curve of a function, referring to the total accumulation of a quantity over time. Both of these processes use limits extensively to reach precise results. Calculus is not only a field of academic interest but also has practical applications in science, engineering, economics, statistics, and many other fields.