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Chapter 1: Problem 6

Finding a limit In Exercises \(5-22,\) find the limit. $$\lim _{x \rightarrow-3} x^{4}$$

Short Answer

Expert verified
The limit of the function as \(x\) approaches -3 is \(81\).

Step by step solution

01

Identify the function and the value \(x\) is approaching

The function given is \(f(x) = x^{4}\) and the value that \(x\) is approaching is -3.
02

Substitute the value into the function

We substitute \(x = -3\) into the function: \(f(-3) = (-3)^{4}\).
03

Simplify the expression

Simplify the expression \(f(-3) = (-3)^{4}\) to get \(81\).

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Key Concepts

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

Continuity
In mathematics, the concept of continuity is vital for understanding limits and functions. A function is continuous at a certain point if you can draw it without lifting your pencil. More formally, a function is continuous at a point if the limit as it approaches that point from either direction equals the function's value at that point. For example, if you approach -3 from either side on the function \( f(x) = x^4 \), you should get the same result. This means that \( \lim_{x \to -3} x^4 = 81 \).
In simple terms:
  • Continuity means no breaks, jumps, or holes at a point.
  • The left-hand limit and right-hand limit must agree.
  • The function鈥檚 value at the point must match the limit value.
Understanding continuity helps in predicting behavior near a given point. This forms the basis of calculus, enabling more advanced problem solving and analysis.
Substitution Method
The substitution method is a straightforward technique used to find limits, especially when dealing with polynomial functions. It involves directly substituting the value that \(x\) approaches into the function. This method hinges on the continuity of the function being evaluated. If a function is continuous, the substitution method will always work smoothly.
Why it works:
  • Polynomials are inherently continuous everywhere on their domain.
  • It allows solving limits by simplification rather than complex algebra.
For our function \(f(x) = x^4\), using the substitution method is simple:1. Substitute \(x = -3\) into \(f(x)\).2. Calculate \((-3)^4\), which simplifies to 81.
This makes it one of the most user-friendly methods for evaluating limits.
Evaluating Polynomials
Evaluating polynomials involves finding a specific output for a given input value. In simpler words, you take the polynomial function and plug a number into it to find the result. This is an essential skill in mathematics because polynomials frequently appear in various types of problems.
Why it鈥檚 useful:
  • Polynomials describe a wide range of natural processes.
  • They are easy to work with due to their continuous and differentiable nature.
  • Evaluating them is straightforward: substitute and compute.
For the function given \(f(x) = x^4\) at \(x = -3\):
  • Substitute \(-3\) into \(x^4\).
  • Calculate \((-3)^4 = 81\).
This approach can also be extended to higher degree polynomials and even polynomials with multiple terms. Knowing how to evaluate polynomials provides a solid foundation for tackling more complex mathematical concepts.

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Most popular questions from this chapter

Proof Prove that if \(f\) is continuous and has no zeros on \([a, b],\) then either \(f(x)>0\) for all \(x\) in \([a, b]\) or \(f(x)<0\) for all \(x\) in \([a, b]\)

Using the Intermediate Value Theorem In Exercises 89-94, use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$g(t)=2 \cos t-3 t$$

Volume Use the Intermediate Value Theorem to show that for all spheres with radii in the interval \([5,8],\) there is one with a volume of 1500 cubic centimeters.

Evaluating Limits Use a graphing utility to evaluate $$\lim _{x \rightarrow 0} \frac{\tan n x}{x}$$ for several values of \(n .\) What do you notice?

Continuity of a Composite Function In Exercises \(65-70\) , discuss the continuity of the composite function $$f(x)=\tan x$$ $$g(x)=\frac{x}{2}$$
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