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Chapter 2: Problem 16

Limits of linear functions Evaluate the following limits. $$\lim _{x \rightarrow-5} \pi$$

Short Answer

Expert verified
Answer: The limit is \(\pi\).

Step by step solution

01

Identify the function type

We are given the constant function \(f(x) = \pi\). Note that this function doesn't depend on the value of \(x\).
02

Evaluate the limit of the constant function

Since the function is constant, its limit as \(x\) approaches any value will be the same constant value. Therefore, the limit of the given function is: $$\lim_{x \rightarrow -5} \pi = \pi$$ So, the limit of the given function as \(x\) approaches \(-5\) is equal to \(\pi\).

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Key Concepts

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

Limits
Limits are a fundamental concept in calculus that describe what value a function approaches as the input nears a particular point. Imagine driving down a road toward a destination. The idea of limits is similar; as you get closer and closer, the landscape you see from the car window is like the value reached by the function.

In mathematical terms, when we write \( \lim_{x \to a} f(x) = L \), we express that as \(x\) gets very close to \(a\), the values of the function \(f(x)\) approach \(L\).

Some important points about limits:
  • Limits help determine behavior at points where a function might not be defined.
  • They are essential for defining derivatives and continuity.
  • Understanding limits leads to a deeper insight into the function's behavior as inputs change.
When dealing with limits, remember that it's the approaching behavior we're interested in鈥攐ften, even if the function isn't defined at the point itself.
Constant functions
Constant functions are special kinds of functions where the output value remains the same, regardless of the input. Picture a horizontal line on a graph; that's a constant function.

The general form of a constant function is \( f(x) = c \), where \(c\) is a constant. This means:
  • The value of \( f(x) \) doesn't change as \( x \) changes.
  • Graphically, it鈥檚 a flat line parallel to the x-axis.
  • In our example, \( f(x) = \pi \), \(\pi\) being a mathematical constant, so wherever \(x\) goes, \(f(x)\) remains \(\pi\).
Constant functions are straightforward to work with when evaluating limits, as they inherently simplify many calculations.
They also pave the way for understanding more complex types of functions you鈥檒l encounter.
Evaluating limits
Evaluating limits involves figuring out what value a function approaches as the input nears a specific point. For constant functions, this process is particularly simple.

Since a constant function doesn鈥檛 change its output, the limit is simply that constant, irrespective of the point approached. Consider the expression \( \lim_{x \to -5} \pi \); you substitute \(\pi\) for any \(x\) and find:
  • This equals \(\pi\) itself, because what's constant stays constant.
  • It doesn鈥檛 matter if you鈥檙e approaching from the left or right; the result stays stuck at \(\pi\).
The elegance of this is what makes calculus manageable鈥攗nderstanding such fundamental properties allows for tackling more advanced topics.
Evaluation doesn't change even if the approach point shifts, reinforcing stability and predictability.

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Most popular questions from this chapter

Let $$g(x)=\left\\{\begin{array}{ll}1 & \text { if } x \geq 0 \\\\-1 & \text { if } x<0\end{array}\right.$$ a. Write a formula for \(|g(x)|\) b. Is \(g\) continuous at \(x=0 ?\) Explain. c. Is \(|g|\) continuous at \(x=0 ?\) Explain. d. For any function \(f,\) if \(|f|\) is continuous at \(a,\) does it necessarily follow that \(f\) is continuous at \(a ?\) Explain.

Find the vertical and horizontal asymptotes of \(f(x)=e^{1 / x}\).

a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{x^{2}-4 x+3}{x-1}$$

Electric field The magnitude of the electric field at a point \(x\) meters from the midpoint of a \(0.1-\mathrm{m}\) line of charge is given by \(E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}\) (in units of newtons per coulomb, \(\mathrm{N} / \mathrm{C}\) ). Evaluate \(\lim _{x \rightarrow 10} E(x)\).

Let \(f(x)=\frac{2 e^{x}+10 e^{-x}}{e^{x}+e^{-x}} .\) Evaluate \(\lim _{x \rightarrow 0} f(x), \lim _{x \rightarrow-\infty} f(x),\) and \(\lim _{x \rightarrow \infty} f(x) .\) Then give the horizontal and vertical asymptotes of \(f\) Plot \(f\) to verify your results.
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