Indeterminate Forms
A corner store in Santiago produces bottled passion fruit juice. It has fixed costs of $\text{RD}\$2{,}000$ and each bottle costs $\text{RD}\$50$ to produce. The average cost per bottle is $C(x)=\dfrac{50x+2000}{x}$. If it produces enormous quantities, what value does the average cost get close to? Substituting $x\to\infty$ gives $\frac{\infty}{\infty}$, which tells you nothing by itself. In this page you will learn how to resolve that form and the others that cannot be settled right away.
The catalog of indeterminate forms
When computing a limit, the first step is to substitute. Sometimes the result is a number or $\pm\infty$ and you are done. Other times a form comes out that does not let you conclude.
When there is NO indeterminate form: $\frac{k}{0}$
If the numerator approaches a number $k\ne0$ and the denominator approaches $0$, the limit is not indeterminate: the function shoots off to $+\infty$ or to $-\infty$. To find out which one, compute the one-sided limits ($x\to a^-$ from the left and $x\to a^+$ from the right). The overall limit exists only if both one-sided limits agree.
In $f(x)=\dfrac{2}{x-4}$, when $x\to4^+$ the denominator is a very small positive number, so $f\to+\infty$. When $x\to4^-$ it is a very small negative number, so $f\to-\infty$. The one-sided limits are different and the overall limit does not exist.
Observe the following graph of the function $f(x)$ and determine, intuitively, the value of the indicated limits. If any limit does not exist or is infinite, state so…
Observe the following graph of the function $f(x)$ and determine, intuitively, the value of the indicated limits. If any limit does not exist or is infinite, state so clearly.
- a) $\lim_{x \to -2} f(x)$
- b) $\lim_{x \to 0} f(x)$
- c) $\lim_{x \to 3} f(x)$
- d) $\lim_{x \to +\infty} f(x)$
Given the rational function $f(x) = \frac{2}{x-4}$, study its behavior near its vertical asymptote by calculating the one-sided limits: a)…
Given the rational function $f(x) = \frac{2}{x-4}$, study its behavior near its vertical asymptote by calculating the one-sided limits:
- a) $\lim_{x \to 4^-} \frac{2}{x-4}$
- b) $\lim_{x \to 4^+} \frac{2}{x-4}$
State with reasons whether the overall limit exists at $x = 4$.
$\frac{0}{0}$: factoring and conjugates
When $x\to a$ and both numerator and denominator equal $0$, both have the factor $(x-a)$ hidden inside. The idea is to bring it out and cancel it.
Calculate the limit of the function $f(x) = \frac{x^2 - 9}{x - 3}$ as $x$ tends to $3$. Help: Notice that direct substitution gives an indeterminate form of the type…
Calculate the limit of the function $f(x) = \frac{x^2 - 9}{x - 3}$ as $x$ tends to $3$.
Help: Notice that direct substitution gives an indeterminate form of the type $\left[\frac{0}{0}\right]$. Try factorising the numerator to simplify the expression.
Calculate the following limit by resolving the indeterminate form $\frac{0}{0}$ using the conjugate technique: $$\lim_{x \to 4} \frac{x^2 - 16}{\sqrt{x+5} - 3}$$
Calculate the following limit by resolving the indeterminate form $\frac{0}{0}$ using the conjugate technique:
$$\lim_{x \to 4} \frac{x^2 - 16}{\sqrt{x+5} - 3}$$Solve the following limit using the double conjugate technique to resolve the $\frac{0}{0}$ indeterminate form: $$\lim_{x \to 0} \frac{\sqrt{1+x} -…
Solve the following limit using the double conjugate technique to resolve the $\frac{0}{0}$ indeterminate form:
$$\lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{4+x} - 2}$$$\frac{\infty}{\infty}$: comparing degrees
If $x\to\pm\infty$ and both numerator and denominator grow without bound, the one that grows faster wins.
Back to the corner store's average cost: the degrees are equal (both $1$) and the leading coefficients are $50$ and $1$. So $\displaystyle\lim_{x\to+\infty}\frac{50x+2000}{x}=50$ pesos per bottle. The fixed costs get diluted when production is large.
Calculate the following limits of polynomial functions as the variable $x$ tends to infinity. Remember to identify the dominant term in each case: a)…
Calculate the following limits of polynomial functions as the variable $x$ tends to infinity. Remember to identify the dominant term in each case:
- a) $\lim_{x \to +\infty} (-2x^3 + 5x^2 - 1)$
- b) $\lim_{x \to -\infty} (x^4 - 100x^2)$
- c) $\lim_{x \to +\infty} \frac{3x^2 - 5}{10}$
Resolve the indeterminate form $\left[\frac{\infty}{\infty}\right]$ in the following limits of rational functions by comparing the degrees of the numerator and…
Resolve the indeterminate form $\left[\frac{\infty}{\infty}\right]$ in the following limits of rational functions by comparing the degrees of the numerator and denominator:
- a) $\lim_{x \to +\infty} \frac{5x^2 - 3x + 2}{2x^2 + 1}$
- b) $\lim_{x \to +\infty} \frac{x + 5}{x^2 - 3}$
- c) $\lim_{x \to +\infty} \frac{x^3 - 1}{x^2 + 10x}$
Determine the value of the limit at infinity by analysing the degree of the dominant terms in the numerator and the denominator. Take special care with the powers inside…
Determine the value of the limit at infinity by analysing the degree of the dominant terms in the numerator and the denominator. Take special care with the powers inside the roots:
$$\lim_{x \to +\infty} \frac{\sqrt{9x^2 + 2x - 1} + 2x}{\sqrt[3]{x^3 + 5x^2}}$$Hierarchy of infinities
If logarithms, powers and exponentials are mixed in the quotient, there are no degrees to compare. You use the order of growth rates.
A radical also counts as a degree: $\sqrt{x^{20}-3x^2}$ behaves like $x^{10}$, and $\sqrt{x^7+1}$ like $x^{3.5}$.
In the study of computer algorithms, it is crucial to know which function grows faster. Based on the hierarchy of infinities, assign the value of the following limits…
In the study of computer algorithms, it is crucial to know which function grows faster. Based on the hierarchy of infinities, assign the value of the following limits ($0$, $+\infty$ or $-\infty$):
- a) $\lim_{x \to +\infty} \frac{\ln(x)}{x}$
- b) $\lim_{x \to +\infty} \frac{e^x}{x^{10}}$
- c) $\lim_{x \to +\infty} \frac{x^2 + 1}{\ln(x)}$
Use comparison of orders of infinity to solve the following limits without performing lengthy calculations. Justify your answer by indicating which function "dominates"…
Use comparison of orders of infinity to solve the following limits without performing lengthy calculations. Justify your answer by indicating which function "dominates" the other:
- a) $\lim_{x \to +\infty} \frac{e^x + x^5}{100x^6 - \ln(x)}$
- b) $\lim_{x \to +\infty} \frac{\sqrt{x^7 + 1}}{x^3 \times \ln(x)}$
Calculate the value of the following limit by analysing the hierarchy of the infinities involved. Give reasons to justify which term dominates in the numerator and in…
Calculate the value of the following limit by analysing the hierarchy of the infinities involved. Give reasons to justify which term dominates in the numerator and in the denominator before providing the final result:
$$\lim_{x \to +\infty} \frac{3 \times e^{2x} - \ln(x^5 + 1) + x^{10}}{2 \times 5^x + \sqrt{x^{20} - 3x^2} - e^x}$$$\infty-\infty$: operating and conjugating
When you subtract two things that grow without bound, the result depends on which one grows faster and on how similar they are.
Resolve the following indeterminate form $\infty - \infty$. To do so, perform the preliminary algebraic operation and simplify the resulting expression: $$\lim_{x \to 1}…
Resolve the following indeterminate form $\infty - \infty$. To do so, perform the preliminary algebraic operation and simplify the resulting expression:
$$\lim_{x \to 1} \left( \frac{6}{x^2 - 1} - \frac{3}{x - 1} \right)$$Calculate the value of the following expression which presents an indeterminate form of the type $\infty - \infty$ with roots of different orders: $$\lim_{x \to +\infty}…
Calculate the value of the following expression which presents an indeterminate form of the type $\infty - \infty$ with roots of different orders:
$$\lim_{x \to +\infty} \left[ \sqrt{x^2 + 4x} - \sqrt[3]{x^3 - 6x^2} \right]$$$1^\infty$ and the number $e$
If the base tends to $1$ and the exponent tends to $\infty$, the result depends on how fast the base approaches $1$. The model case is
$$\lim_{x\to+\infty}\left(1+\frac1x\right)^{x}=e\approx2.718.$$Calculate the value of the following limit leading to an indeterminate form of the type $[1^\infty]$ using the shortcut formula for the number $e$: $$\lim_{x \to…
Calculate the value of the following limit leading to an indeterminate form of the type $[1^\infty]$ using the shortcut formula for the number $e$:
$$\lim_{x \to +\infty} \left( 1 + \frac{1}{x} \right)^{3x}$$In the development of a new social network, the number of active users (in millions) follows the function $U(t) = \left(\frac{2t+3}{2t-1}\right)^{t+1}$, where $t$ is the…
In the development of a new social network, the number of active users (in millions) follows the function $U(t) = \left(\frac{2t+3}{2t-1}\right)^{t+1}$, where $t$ is the time in months since its launch (with $t>\frac{1}{2}$). Calculate the value the number of users approaches as time grows indefinitely ($t \to +\infty$).
In the design of a new telecommunications antenna, the signal intensity $I$ as a function of the distance $d$ to the repeater is given by the expression: $$I(d) = \left(…
In the design of a new telecommunications antenna, the signal intensity $I$ as a function of the distance $d$ to the repeater is given by the expression:
$$I(d) = \left( \frac{d^2 + 5d}{d^2 - 2d + 1} \right)^{2d + 3}$$Calculate the limiting signal intensity as distance $d$ tends to infinity (indeterminate form of the type $1^\infty$).
Parameters, one-sided limits and piecewise functions
There are exercises where the limit is given and what you are looking for are the parameters. The strategy is: force the indeterminate form to be resolvable and then set it equal to the required value.
Piecewise functions and one-sided limits
For the limit at a point to exist, the two one-sided limits must be equal. If each piece becomes $\frac00$, resolve each side separately. Two useful reminders:
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$|x-1|=x-1$ if $x>1$ and $=-(x-1)$ if $x<1$. When computing one-sided limits, remove the absolute value according to the side.
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$\displaystyle\lim_{u\to0}\frac{\sin u}{u}=1$. That is why $\dfrac{\sin(x-1)}{x-1}\to1$ as $x\to1$.
For $e^{1/(x-1)}$: if $x\to1^+$, the exponent $\to+\infty$ and $e^{1/(x-1)}\to+\infty$. If $x\to1^-$, the exponent $\to-\infty$ and $e^{1/(x-1)}\to0$.
Find the values of the real parameters $a$ and $b$ so that the following limit is finite and equals $4$: $$\lim_{x \to +\infty} \sqrt{x^2 + ax + 1} - \sqrt{x^2 + bx}$$
Find the values of the real parameters $a$ and $b$ so that the following limit is finite and equals $4$:
$$\lim_{x \to +\infty} \sqrt{x^2 + ax + 1} - \sqrt{x^2 + bx}$$Given the following graph of the function $f(x)$, analyse visually and calculate analytically the one-sided limits at the points of discontinuity to determine whether…
Given the following graph of the function $f(x)$, analyse visually and calculate analytically the one-sided limits at the points of discontinuity to determine whether the overall limit exists at $x = 2$:
$$f(x) = \begin{cases} \frac{x^2 - 4}{x - 2} & \text{if } x < 2 \\ ax - 1 & \text{if } x \geq 2 \end{cases}$$Calculate the value of $a$ so that the limit at $x = 2$ is unique and the function does not have a jump.
Determine the values of the real parameters $a$ and $b$ so that the following equality holds: $$\lim_{x \to 2} \frac{\sqrt{x^2 + ax + b} - 2}{x^2 - 5x + 6} =…
Determine the values of the real parameters $a$ and $b$ so that the following equality holds:
$$\lim_{x \to 2} \frac{\sqrt{x^2 + ax + b} - 2}{x^2 - 5x + 6} = -\frac{1}{2}$$That is, the limit must be a finite real number and equal $-\dfrac{1}{2}$. Once you have found $a$ and $b$, check that the limit really equals $-\dfrac{1}{2}$.