At what points are the functions in Exercises 13–30 continuous?

y = cos (x) / x

Verified step by step guidance

Step 1: Understand the concept of continuity. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.

Step 2: Identify the points where the function might be discontinuous. The function y = cos(x)/x is a rational function, which is typically continuous except where the denominator is zero.

Step 3: Determine where the denominator is zero. Set the denominator x equal to zero and solve for x. In this case, x = 0.

Step 4: Analyze the behavior of the function at x = 0. Since the denominator is zero at x = 0, the function is not defined at this point, indicating a potential discontinuity.

Step 5: Conclude the points of continuity. The function y = cos(x)/x is continuous for all x except at x = 0, where it is undefined.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. This means there are no breaks, jumps, or holes in the graph of the function at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are crucial for determining continuity, especially at points where the function may not be explicitly defined. For the function y = cos(x) / x, evaluating the limit as x approaches 0 is essential to assess its continuity at that point.

Undefined Points

A function can be undefined at certain points, which can affect its continuity. In the case of y = cos(x) / x, the function is undefined at x = 0 because division by zero is not permissible. Understanding where a function is undefined helps identify potential discontinuities and informs the analysis of its overall continuity.

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Textbook Question

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

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Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (tan 2x) / x

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Textbook Question

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim x → 0 tan (π/4 cos (sin x¹/³))