Selecting Procedures for Determining Limits
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AP Calculus AB › Selecting Procedures for Determining Limits
Direct substitution into the limit $$ \lim_{x \to 0} \frac{e^x - 1}{\sin(x)} $$ yields $$ \frac{0}{0} $$. Which procedure is most effective for evaluating this limit without using L'Hôpital's Rule?
Factor $$e^x$$ from the numerator and use trigonometric identities to simplify the denominator.
Apply the Squeeze Theorem by bounding the function between two simpler functions that approach the same value.
Multiply the numerator and denominator by the conjugate of the numerator, $$e^x + 1$$, to simplify the expression.
Multiply the numerator and denominator by $$x$$ and rearrange to use the known limits for $$ \frac{e^x-1}{x} $$ and $$ \frac{x}{\sin(x)} $$.
Explanation
This limit can be resolved by using two known fundamental limits: $$ \lim_{x \to 0} \frac{e^x - 1}{x} = 1 $$ and $$ \lim_{x \to 0} \frac{\sin x}{x} = 1 $$. The expression can be rewritten as $$ \lim_{x \to 0} \frac{(e^x - 1)/x}{\sin(x)/x} $$. Using the quotient property of limits, the limit is $$ \frac{\lim_{x \to 0} \frac{e^x-1}{x}}{\lim_{x \to 0} \frac{\sin x}{x}} = \frac{1}{1} = 1 $$.
Which of the following is the most appropriate method to find the limit $$ \lim_{x \to \infty} \frac{\sqrt{9x^2 + x}}{2x+1} $$?
Multiply the numerator and denominator by the conjugate of the numerator, $$ \sqrt{9x^2+x} $$.
Use direct substitution, which is the first step for all limit problems and will give the answer here.
Apply the Squeeze Theorem, since the function contains a square root and is difficult to evaluate directly.
Divide the numerator and denominator by $$x$$, remembering that for $$x>0$$, $$x = \sqrt{x^2}$$.
Explanation
For a limit at infinity involving a rational-like expression with a radical, the most effective method is to divide the numerator and denominator by the highest power of $$x$$ in the denominator, which is $$x$$. To divide the numerator by $$x$$, we use the fact that $$x = \sqrt{x^2}$$ for positive $$x$$ to bring the term inside the square root. This simplifies the expression and allows for evaluation of the limit.
To evaluate the limit $$ \lim_{h \to 0} \frac{\frac{1}{4+h} - \frac{1}{4}}{h} $$, which of the following algebraic procedures should be performed first?
Cancel the term $$h$$ from the numerator and the denominator immediately.
Multiply the numerator and denominator by the conjugate of the numerator.
Combine the fractions in the numerator into a single fraction using a common denominator.
Apply direct substitution to the expression as it is written to find the final value.
Explanation
Direct substitution results in the indeterminate form $$ \frac{0}{0} $$. The expression contains a complex fraction in the numerator. The correct first step is to simplify this by finding a common denominator for the terms $$ \frac{1}{4+h} $$ and $$ \frac{1}{4} $$. This will result in a single fraction in the numerator, which can then be simplified further, allowing for cancellation of the $$h$$ term in the denominator.
Which of the following statements describes the most appropriate procedure for determining the limit $$ \lim_{x \to 1} \frac{x-1}{|x-1|} $$?
Because direct substitution results in $$ \frac{0}{0} $$, one should simplify the expression by factoring.
Since the expression contains an absolute value, the Squeeze Theorem must be used to find the limit.
The limit must be evaluated by considering the limits from the left and right, as the definition of $$|x-1|$$ changes at $$x=1$$.
The expression can be simplified by multiplying the numerator and denominator by the conjugate of the denominator.
Explanation
The absolute value function $$|x-1|$$ is defined piecewise: $$|x-1| = x-1$$ for $$x > 1$$ and $$|x-1| = -(x-1)$$ for $$x < 1$$. Because the rule changes at $$x=1$$, it is essential to evaluate the one-sided limits. The limit from the right is 1, and the limit from the left is -1. Since they differ, the two-sided limit does not exist.
To determine $$ \lim_{x \to 2} f(x) $$, which of the following procedures is required?
Evaluating $$f(2)$$ directly since the function is defined at $$x=2$$ and using that value as the limit.
Evaluating the limit from the left ($$x \to 2^-$$) and the limit from the right ($$x \to 2^+$$) and comparing them.
Applying the Squeeze Theorem because the function is defined by two different rules on each side of $$x=2$$.
Factoring the quadratic and linear expressions and canceling common terms before taking the limit.
Explanation
Because the function's definition changes at the point $$x=2$$, it is necessary to determine if the limit from the left is equal to the limit from the right. One must calculate $$ \lim_{x \to 2^-} f(x) $$ using $$x^2+1$$ and $$ \lim_{x \to 2^+} f(x) $$ using $$3x-1$$. The two-sided limit exists if and only if these one-sided limits are equal.
Which of the following is the most effective method for determining the limit $$ \lim_{x \to \infty} \frac{3x^3 - 2x + 1}{5x^3 + x^2 - 7} $$?
Applying the Squeeze Theorem by comparing the function to simpler bounding functions.
Multiplying the numerator and denominator by the conjugate of the denominator.
Factoring the numerator and denominator to cancel common polynomial factors.
Dividing every term in the numerator and denominator by $$x^3$$, the highest power of $$x$$ in the denominator.
Explanation
For limits at infinity of rational functions, the standard and most effective procedure is to analyze the end behavior by dividing both the numerator and the denominator by the highest power of $$x$$ that appears in the denominator. This transforms the expression into a form where the limits of individual terms can be easily evaluated as $$x$$ approaches infinity.
To evaluate the limit $$ \lim_{x \to 3} \frac{x^2 - 9}{x^2 - 2x - 3} $$, direct substitution results in the indeterminate form $$ \frac{0}{0} $$. Which of the following is the most appropriate algebraic method to use next?
Multiplying the numerator and denominator by the conjugate of the numerator, $$x^2+9$$.
Applying the special trigonometric limit $$ \lim_{x \to 0} \frac{\sin x}{x} = 1 $$ after a substitution.
Using the Squeeze Theorem with functions that bound the given expression near $$x=3$$.
Factoring the expressions in the numerator and denominator and simplifying the resulting rational expression.
Explanation
The expression is a rational function. For the indeterminate form $$ \frac{0}{0} $$, the appropriate procedure is to factor the numerator into $$ (x-3)(x+3) $$ and the denominator into $$ (x-3)(x+1) $$. This allows for cancellation of the $$ (x-3) $$ term, which resolves the indeterminate form and allows for evaluation by direct substitution into the simplified expression.
To analyze the limit of the function $$g(x)$$ as $$x$$ approaches 3, which procedure is necessary?
Conclude the limit is 5 because the function is explicitly defined as $$g(3)=5$$.
Factor the denominator and simplify the expression before applying direct substitution.
Evaluate the one-sided limits as $$x \to 3^-$$ and $$x \to 3^+$$ to determine if the function approaches $$ \infty $$ or $$ -\infty $$.
Use direct substitution into the expression $$ \frac{1}{x-3} $$ because it defines the function near $$x=3$$.
Explanation
The limit of a function at a point depends on the values of the function near that point, not at the point. Near $$x=3$$, the function is defined by $$ \frac{1}{x-3} $$. Substituting $$x=3$$ into this expression gives the form $$ \frac{1}{0} $$, which indicates a vertical asymptote. To fully describe the behavior, it is necessary to examine the one-sided limits. The value $$g(3)=5$$ is irrelevant to the value of the limit.
To evaluate the limit $$ \lim_{x \to 0} \frac{\sqrt{x+9} - 3}{x} $$, evaluating using direct substitution results in an indeterminate form. Which of the following is the most appropriate next step?
Multiplying the numerator and denominator by the conjugate expression $$ \sqrt{x+9} + 3 $$.
Applying the Squeeze Theorem by bounding the function between $$y=-1/x$$ and $$y=1/x$$.
Factoring a common term of $$x$$ from the numerator and denominator.
Simplifying the expression by finding a common denominator for terms in the numerator.
Explanation
The limit results in the indeterminate form $$ \frac{0}{0} $$. The presence of a square root in the numerator suggests that multiplying the numerator and denominator by its conjugate, $$ \sqrt{x+9} + 3 $$, is the most effective procedure. This step removes the radical from the numerator and creates a term that can be cancelled with the denominator.
Which of the following procedures should be used to evaluate the limit $$ \lim_{x \to \pi/2} \frac{\sin(x)}{x} $$?
The Squeeze Theorem, because the function involves a trigonometric component which is bounded.
Multiplying by a conjugate, which is a standard method for resolving indeterminate forms.
Direct substitution, because the function $$f(x) = \frac{\sin(x)}{x}$$ is continuous at $$x=\pi/2$$.
Algebraic simplification by factoring, because the expression is a rational function of trigonometric terms.
Explanation
The function $$f(x) = \frac{\sin(x)}{x}$$ is continuous for all $$x \neq 0$$. Since the limit is being evaluated at $$x=\pi/2$$, a point within the domain of continuity, direct substitution is the correct and simplest method. The result is $$ \frac{\sin(\pi/2)}{\pi/2} = \frac{1}{\pi/2} = \frac{2}{\pi} $$. No other procedure is necessary.