Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 


f(x)=1+sin x / cos x; limx→π/2^− f(x); lim x→4π/3 f(x)

Verified step by step guidance

1. <Step 1: Identify the function and its components.> The function given is $f(x)=1+\frac{\sin x}{\cos x}$. This can be rewritten as $f(x)=1+\tan x$ since $\tan x=\frac{\sin x}{\cos x}$.
2. <Step 2: Determine the continuity of $\tan x$.> The function $\tan x$ is continuous wherever $\cos x\ne 0$. The cosine function is zero at odd multiples of $\frac{\pi }{2}$, i.e., $x=\frac{\pi }{2}+k\pi$ where $k$ is an integer.
3. <Step 3: Establish the intervals of continuity for $f(x)$.> Since $f(x)=1+\tan x$, it is continuous on intervals where $\tan x$ is continuous. Therefore, $f(x)$ is continuous on intervals of the form $(k\pi -\frac{\pi }{2},k\pi +\frac{\pi }{2})$ for integer $k$.
4. <Step 4: Analyze the limit $\underset{x\to \pi /2^{-}}{lim}f(x)$.> As $x$ approaches $\pi /2$ from the left, $\tan x\to +\mathrm{\infty }$ because $\cos x\to 0^{+}$. Therefore, $f(x)=1+\tan x\to +\mathrm{\infty }$.
5. <Step 5: Analyze the limit $\underset{x\to 4\pi /3}{lim}f(x)$.> At $x=4\pi /3$, $\cos (4\pi /3)\ne 0$, so $\tan (4\pi /3)$ is defined. Calculate $\tan (4\pi /3)$ and substitute into $f(x)=1+\tan x$ to find the limit.>

Key Concepts

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. Understanding continuity is essential for analyzing the behavior of functions, especially when determining where they are defined and how they behave near critical points.

Limits

A limit describes the value that a function approaches as the input approaches a certain point. Limits are fundamental in calculus, particularly for understanding the behavior of functions at points where they may not be explicitly defined, such as points of discontinuity. Evaluating limits helps in analyzing the continuity and differentiability of functions, which is crucial for solving problems involving calculus.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. They play a significant role in calculus, especially in determining the behavior of functions involving angles. Understanding the properties of these functions, including their continuity and limits, is vital for analyzing expressions like f(x) = (1 + sin x) / cos x, particularly at points where the denominator may approach zero.