2.2.4How do algebraic manipulations help determine a limit?

Evaluating Limits

So far you have looked at limits on graphs and in terms of continuity. Today you will apply algebraic computations to determine limits and better understand graphs.

2-89.

A NEW REASON TO SIMPLIFY

Yashar was trying to determine the following limit: $\lim\limits_{ x \rightarrow 2 } \frac { x ^ { 2 } - 2 x } { x ^ { 2 } - 4 }$ . When he substituted in $x = 2$ he got $\frac { 0 } { 0 }$ . Explain why Yashar cannot determine this limit in its current form.
Meanwhile Hripsime rewrote the limit as $\lim\limits_{ x \rightarrow 2 } \frac { x } { x + 2 }$ . Explain what is happening on the graph at $x = 2$ .
Determine the value of the limit in part (b).
Explain why Hripsime’s method is useful.

2-90.

Given the limits below, state the algebraic method(s) that can help you evaluate the limit.

$\lim\limits_ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { x - 3 }$

$\lim\limits_ { x \rightarrow 3 ^ { + } } \frac { 1 } { x - 3 }$

$\lim\limits_ { x \rightarrow 9 } \frac { x - 9 } { \sqrt { x } - 3 }$

Which problem(s) above have limits that exists and which problem(s) have limits that do not exist? Explain the graphical significance of your answer.

2-91.

Evaluate the following limits.

$\lim\limits_ { x \rightarrow 1 } \frac { 2 x ^ { 2 } - 2 } { x - 1 }$

$\lim\limits_ { x \rightarrow \infty } \frac { 2 x ^ { 2 } - 7 } { x ^ { 2 } + 4 x - 1 }$

$\lim\limits_ { x \rightarrow 0 } \frac { 2 x ^ { 3 } - 7 } { 3 x ^ { 3 } + 4 x - 5 }$

$\lim\limits_ { x \rightarrow 1 } \frac { \sqrt { x } - 1 } { x - 1 }$

$\lim\limits_ { x \rightarrow 4 ^ { - } } \frac { 5 } { x - 4 }$

$\lim\limits_ { x \rightarrow - \infty } \frac { x ^ { 2 } - 1 } { x + 2 }$

2-92.

For each function below, complete the following tasks:

List all horizontal asymptotes, if any, then determine $\lim\limits_ { x \rightarrow\infty } f ( x )$ and $\lim\limits_ { x \rightarrow - \infty } f ( x )$ .
List all vertical asymptotes and holes, if any, then determine $\lim\limits_ { x \rightarrow V . A^+ } f ( x )$ , $\lim\limits_ { x \rightarrow V . A ^ { - } } f ( x )$ , and $\lim\limits_ { x \rightarrow \text { holes } } f ( x )$ .

$f(x) = \frac { 2 ( x - 3 ) } { 3 x - 15 }$

$f(x) = \text{arctan}(x)$

$f(x) = \frac { 4 ( x - 2 ) ( x - 3 ) } { x - 2 }$

2-93.

In CPM Precalculus, limits were used to compare the magnitudes of various functions as they “raced” to infinity. It was discovered that as exponential, power, and logarithmic functions approached infinity an order of domination emerged in the race. The results are summarized below showing the order of domination.

$\text{logarithmic}<\text{power}<\text{exponential}$

Within each function family there exists an order of domination as well. For example, as $x$ approaches infinity:

Exponential:      $1.1^x < 2^x < 8^x$
Power:                $\sqrt { x }< x^2 < x^{10}$
Logarithmic:      $\log_{10}(x)<\log_5(x)<\log_2(x)$

Use the idea of dominant terms to evaluate the following limits:

$\lim\limits_ { x \rightarrow \infty } \frac { 2 ^ { x } + x ^ { 3 } } { x ^ { 5 } + x }$

$\lim\limits_ { x \rightarrow \infty } \frac { 2 x ^ { 2 } - 8 x } { 5 x ^ { 2 } + x }$

$\lim\limits_ { x \rightarrow \infty } \frac { 2 x ^ { 2 } - 8 x } { 5 x ^ { 2 } + 3 ^ { x } }$

Evaluate each of the limits in parts (a), (b), and (c) again, but this time let $x \rightarrow –\infty$ .

2-94.

Use the graph of $y = f (x)$ below to determine the following values. If the limit does not exist, explain why.

$f (−4)$

$\lim\limits_ { x \rightarrow - 4 } f ( x )$

$f (−1)$

$\lim\limits_ { x \rightarrow - 1 ^ { - } } f ( x )$

$\lim\limits_{ x \rightarrow - 1 ^ { + } } f ( x )$

$\lim\limits_ { x \rightarrow - 1 } f ( x )$

$f (2)$

$\lim\limits_ { x \rightarrow 2 } f ( x )$

$f (4)$

$\lim\limits_ { x \rightarrow 4 } f ( x )$

Is the function continuous at $x = 4$ ? Explain your reasoning using the formal definition of continuity.

2-95.

Inscribed rectangles are below a curve. Circumscribed rectangles are above a curve. For the function $y = \sqrt { 4 - x ^ { 2 } }$ , complete the following problems. 2-95 HW eTool (Desmos). Homework Help ✎

Calculate the area under the curve from $−2 ≤ x ≤ 2$ using four inscribed rectangles.
Calculate the area under the curve from $−2 ≤ x ≤ 2$ using four circumscribed rectangles.
Estimate the actual area under the curve using your answers to parts (a) and (b).

2-96.

Suppose $f$ and $g$ are both discontinuous at $x = 3$ . Using the table below, for which of the functions does the limit as $x$ approaches $3$ appear to exist? Justify your answer. Homework Help ✎

$x$	$2.8$	$2.9$	$2.99$	$3$	$3.01$	$3.1$	$3.2$
$f(x)$	$6.97$	$6.98$	$6.99$	$?$	$7.01$	$7.02$	$7.03$
$g(x)$	$6.97$	$6.98$	$6.99$	$?$	$7.99$	$7.98$	$7.97$

2-97.

Let $f$ be an even function such that $f(2) = 4$ and $f(10) = 20$ . Which of the following statements must be true? Could be true? Must be false? Homework Help ✎

$f\left(-10\right)=20$

$f(-2) = -4$

$f(0) = 0$

2-98.

If $1 < a < b$ , which of the following logarithmic expressions represents a value that is negative? Between $0$ and $1$ ? Equal to $1$ ? Greater than $1$ ? Homework Help ✎

$\log_a(b)$

$\log_b\frac{1}{a}$

$\log_b(a)$

$\log_a(a)$

2-99.

What are the $x$ - and $y$ -intercepts of the graph of $x+3=3^{3(y+1)}$ ? Homework Help ✎

2-100.

Let $f(x) = x^2 − 9$ , and $g(x) = 2x^2 − 12x + 18$ . State all horizontal asymptotes, vertical asymptotes, and holes (if any) for $y = \frac { f ( x ) } { g ( x ) }$ and $y = \frac { g ( x ) } { f ( x ) }$ . Homework Help ✎

2-101.

The region bounded by $y = -x + 6$ and the coordinate axes is rotated about the $y$ -axis. Calculate the volume of the resulting solid. Homework Help ✎

Calculus Third Edition