BAC2 tech · Mathematics

Limits and continuity · Limits

Indeterminate forms

You already know the individual tools: factorising, the conjugate, infinitesimals, the number $e$. This section organises them into a closed list of indeterminate forms and trains the cases where two techniques are combined: roots of different index, parameters that must give a specific result, and exponents that hide a $1^\infty$ with variables.

Before you start, make sure you know: Calculating limits of functions (1st Bachillerato)

Before you start, make sure you know: Radicals (1st Bachillerato)

The complete list of indeterminate forms

When you substitute $x=a$ (or $x\to\infty$) into a limit, there is only a closed set of results that do not let you conclude anything without first transforming the expression.

The indeterminate forms in limit calculations

$$\frac{0}{0} \qquad \frac{\infty}{\infty} \qquad \infty-\infty \qquad 1^{\infty} \qquad 0\cdot\infty \qquad 0^{0} \qquad \infty^{0}$$

The last three reduce to the previous ones: $0\cdot\infty$ is rewritten as a quotient ($f\cdot g=\frac{f}{1/g}$), and $0^0$ and $\infty^0$ are handled by taking logarithms, which turns them into $0\cdot\infty$. Any other result of direct substitution does settle the limit without any further work. For example: a quotient with a non-zero denominator, $k/0$ with $k\ne0$, $\infty+\infty$, $0^{\infty}$, or a positive base different from $1$ raised to a finite or infinite exponent.

Why? They are called indeterminate forms because two opposing tendencies come into conflict (for example, a base that tends to $1$ and an exponent that tends to infinity), so the final result depends on the relative rate at which each factor changes.

Practice★☆☆

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) $\lim_{x \to +\infty} f(x)$

Solve

$\dfrac{\infty}{\infty}$: comparing degrees and the hierarchy

When the numerator and denominator both tend to infinity, the result depends on which one "wins": compare degrees for polynomials, or use the hierarchy of infinities if exponentials, logarithms or factorials are mixed in.

Practice★☆☆

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) $\lim_{x \to +\infty} \frac{x^3 - 1}{x^2 + 10x}$

Solve

Practice★★☆

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:

$$\lim_{x \to +\infty} \frac{\sqrt{9x^2 + 2x - 1} + 2x}{\sqrt[3]{x^3 + 5x^2}}$$

Solve

Practice★★☆

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) $\lim_{x \to +\infty} \frac{\sqrt{x^7 + 1}}{x^3 \cdot \ln(x)}$

Solve

$\infty-\infty$ with roots of different index: the double conjugate

If the subtraction that gives $\infty-\infty$ mixes a square root and a cube root (or any pair of different indices), a single conjugate is not enough: you have to apply the technique twice, once for each type of root.

Practice★★★

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:

$$\lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{4+x} - 2}$$

Solve

Practice★★★

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:

$$\lim_{x \to +\infty} \left[ \sqrt{x^2 + 4x} - \sqrt[3]{x^3 - 6x^2} \right]$$

Solve

Practice★★☆

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:

$$\lim_{x \to 1} \left( \frac{6}{x^2 - 1} - \frac{3}{x - 1} \right)$$

Solve

$1^\infty$ with parameters

When the base and the exponent of a $1^\infty$ depend on a parameter, you first identify the indeterminate form as usual, and then use the formula for the number $e$, keeping the parameter inside the calculation of $L$.

Practice★★☆

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 ($t \to +\infty$).

Solve

Practice★★☆

Calculate the value of the following exponential limit which leads to an indeterminate form of the type $1^\infty$. Express the result in terms of the number $e$:…

Calculate the value of the following exponential limit which leads to an indeterminate form of the type $1^\infty$. Express the result in terms of the number $e$:

$$\lim_{x \to 0} (1 + \sin(x))^{\frac{2}{x}}$$

Solve

Practice★☆☆

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 $e$:

$$\lim_{x \to +\infty} \left( 1 + \frac{1}{x} \right)^{3x}$$

Solve

Parameters so that a limit is finite and non-zero

The most complete exercise in this list: finding $a$ and $b$ so that an indeterminate limit gives a result that is both finite and non-zero. Normally you need to impose two conditions.

Find $a,b\in\mathbb{R}$ such that $\displaystyle\lim_{x\to+\infty} \left(\sqrt{x^2+ax+3}-bx\right) = 4$

If $b\le0$ there is no indeterminate form: the expression tends to $+\infty$. So $b>0$ and $\infty-\infty$ appears. Multiply and divide by the conjugate: the numerator becomes $x^2+ax+3-b^2x^2 = (1-b^2)x^2 + ax + 3$, and the denominator is $\sqrt{x^2+ax+3}+bx$, which behaves like $(1+b)x$ for large $x$.

First condition (surplus degree $=0$): if $1-b^2\ne0$, the numerator has degree $2$ and the denominator degree $1$, and the limit would be infinite. Therefore $1-b^2=0$, that is, $b=1$ (since $b>0$). With $b=1$ we get $\dfrac{ax+3}{\sqrt{x^2+ax+3}+x}$, whose limit is $\dfrac{a}{2}$.

Second condition (the quotient gives the required value, which is non-zero): $\dfrac{a}{2}=4 \Rightarrow a=8$. The two conditions fix the parameters completely: $a=8$, $b=1$.

Practice★★★

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 $-\dfrac{1}{2}$.

Solve

Practice★★☆

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 $4$:

$$\lim_{x \to +\infty} \sqrt{x^2 + ax + 1} - \sqrt{x^2 + bx}$$

Solve

Summary

Closed list of indeterminate forms: $\frac00$, $\frac{\infty}{\infty}$, $\infty-\infty$, $1^\infty$, and also $0\cdot\infty$, $0^0$, $\infty^0$, which reduce to the previous ones. None of them can be solved by substituting without transforming first.
$\frac{\infty}{\infty}$: compare degrees (polynomials) or use the hierarchy of infinities (mixtures with exponentials or logarithms). Beware of $\sqrt{x^2}=|x|$.
$\infty-\infty$ with roots of different index: introduce an intermediate term and apply the appropriate conjugate to each index separately.
$1^\infty$ with parameters: use $L=\lim h(x)\big(g(x)-1\big)$, keeping the parameter inside the calculation; the final limit is $e^L$.
Finding parameters for a finite, non-zero limit usually requires two conditions: the "surplus" degree must vanish and the resulting quotient must not be zero.

Let's practice! 10 medium exercises in this section

★★☆ Calculate the following limit by resolving the indeterminate form $\frac{0}{0}$ using the…★★☆ Determine the value of the limit at infinity by analysing the degree of the dominant…★★☆ Resolve the following indeterminate form $\infty - \infty$. To do so, perform the…★★☆ In the development of a new social network, the number of active users (in millions)…

Let's practice! 10 hard exercises in this section

★★★ Calculate the value of the following limit by analysing the hierarchy of the infinities…★★★ Determine the values of the real parameters $a$ and $b$ so that the following equality…★★★ In the design of a new telecommunications antenna, the signal intensity $I$ as a function…★★★ Solve the following limit using the double conjugate technique to resolve the…

See all the exercises in this section