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Chapter 2: Problem 22

Use the precise definition of a limit to prove the following limits. $$\lim _{x \rightarrow 3} \frac{x^{2}-7 x+12}{x-3}=-1$$

Short Answer

Expert verified
Question: Find the limit of the given function: $$\lim _{x \rightarrow 3} \frac{x^{2}-7 x+12}{x-3}$$ Answer: The limit of the given function as x approaches 3 is -1.

Step by step solution

01

Factor the numerator

To factor the numerator, look for two numbers that multiply to 12 and add up to -7. The two numbers that work are -3 and -4. Rewrite the expression as: $$\frac{(x-3)(x-4)}{x-3}$$
02

Cancel out common terms

Notice that both the numerator and the denominator have a \((x-3)\) term. We can cancel out these terms: $$\frac{(x-3)(x-4)}{x-3}=x-4$$
03

Evaluate the limit using the precise definition

Now that we have canceled out the common terms, we can substitute \(x=3\) into the simplified expression, which is \(x-4\): $$\lim _{x \rightarrow 3}(x-4)=3-4=-1$$ The precise definition of the limit has been used to prove that: $$\lim _{x \rightarrow 3} \frac{x^{2}-7 x+12}{x-3}=-1$$

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Most popular questions from this chapter

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.

Limits with a parameter Let \(f(x)=\frac{x^{2}-7 x+12}{x-a}\) a. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)\) equal a finite number? b. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)=\infty ?\) c. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)=-\infty ?\)

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$g(\theta)=\tan \left(\frac{\pi \theta}{10}\right)$$

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$g(x)=e^{1 / x}$$

a. Sketch the graph of a function that is not continuous at \(1,\) but is defined at 1. b. Sketch the graph of a function that is not continuous at \(1,\) but has a limit at 1.
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