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Chapter 2: Problem 23

Evaluate the limits that exist. $$\lim _{x \rightarrow 1} \frac{x^{2}-x-6}{(x+2)^{2}}$$

Short Answer

Expert verified
The limit as x approaches 1 for the given expression is -2/3.

Step by step solution

01

Factorize the expression

The numerator can be factorized into two binomials. The equation of the numerator \(x^{2}-x-6 = 0\) can be simplified using the rule \((a-b)(a+b) = a^{2}-b^{2}\). Let's search for two numbers that multiplied equals to -6 (c) and added equals to -1 (b), according to the quadratic formula \(x^{2}+bx+c\). The numbers are -3 and 2. Thus, \(x^{2}-x-6\) factored is \((x -3)(x+2)\).
02

Simplify the expression

Now, rewrite the original limit using the factorization: \(\lim_{x \rightarrow 1} \frac{(x -3)(x+2)}{(x+2)^{2}}\). The terms \(x+2\) in the numerator and the denominator can be simplified, which results in \(\lim_{x \rightarrow 1} \frac{x -3}{x+2}\).
03

Solve the limit

Now that there are no undefined results, we can solve the limit by substituting x=1: \(\lim_{x \rightarrow 1} \frac{x -3}{x+2} = \frac{1 -3}{1+2} = -2/3\).

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Key Concepts

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

Factorization
Factorization is a crucial skill in algebra that helps simplify complex expressions. In this particular limit problem, we are given a quadratic expression in the numerator: \[ x^{2}-x-6 \]. To factor this, we need to find two numbers that multiply to \(-6\)and add to \(-1\). These numbers are \(-3\) and \(2\). Thus, we can rewrite the expression as \((x - 3)(x + 2)\). This transformation makes the next steps much more manageable and is part of what makes factorization such a powerful algebraic tool.
  • Helps reveal common factors.
  • Simplifies further calculations.
  • Makes solving equations more straightforward.
Once factorized, the expression can often be simplified, leading us to evaluate limits more easily.
Simplifying Expressions
Once we've factored an expression, the next step is often simplification. Simplifying expressions is essential in mathematics to make calculations easier and results clearer. In our exercise, after factorizing the numerator, the expression becomes: \[ \frac{(x-3)(x+2)}{(x+2)^2} \]. Notice how \((x+2)\) appears in both the numerator and the denominator. This allows us to cancel out one \((x+2)\) term, leading us to a simplified form: \[ \frac{x-3}{x+2} \]. This results in a much simpler fraction to work with when applying limit evaluation techniques. Simplification:
  • Makes calculations direct and less prone to errors.
  • Reduces the problem to a more manageable size.
  • Helps in identifying undefined points and potential solutions.
Substitution Method
In limit evaluation, substitution is often the final step after simplifying the expression. Once we have a simplified form free of indeterminate forms like \(\frac{0}{0}\), we can directly substitute the limit point into the expression to find the limit. For our exercise, we substitute \(x = 1\) into \[ \frac{x-3}{x+2} \], giving: \( \frac{1-3}{1+2} = \frac{-2}{3} \). The substitution method is a straightforward calculation and is applicable when the function is continuous at the point after simplifying:
  • Direct substitution is possible when no division by zero occurs post simplification.
  • Provides the ultimate value of the limit.
  • Saves time and reduces complexity in evaluation.
Substitution confirms whether our algebraic manipulations led us to the correct limit value.

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Most popular questions from this chapter

Use the limit utility in a \(C A S\) evaluate the limit. Use a graphing utility to plot \(f(x)=\frac{x}{\tan 3 x}\) on \([-0.2 .0 .2]\) Estimate \(\lim _{x \rightarrow 0} f(x) ;\) use the zoom function if necessary. Verify your result analytically.

After estimating the limit using the prescribed values of \(x,\) validate or improve your estimate by using a graphing utility. Estimate $$\lim _{x \rightarrow 1} \frac{\tan 2 x}{x} \quad \text { (radian meusure) }$$ by evaluating the quotient at \(x=\pm 1 . ; 0.1, 0.01,10.001\)

Suppose that \(f\) has an essential discontinuity at \(c .\) Change the value of \(f\) as you choose at any finite number of points \(x_{1}, x_{2} \ldots \ldots x_{n}\) and call the resulting function \(g .\) Show that \(g\) also has an essential discontinuity at \(c\).

Evaluate the limits that exist. $$\lim _{x \rightarrow 0} \frac{1}{2 x \csc x}$$

Use a graphing utility to graph / on the given interval. Is \(\int\) bounded? Does it have extreme values? If so, what are these extreme values? $$f(x)=\frac{2 x}{1+x^{2}} ; \quad[-2.2]$$
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