Calculating Limits of Functions
Ximena wants to know what value $\dfrac{x^2-25}{x-5}$ approaches as $x$ approaches $5$. She types $x=5$ into her calculator and it shows "Error": she got $\frac{0}{0}$. Does that mean the limit doesn't exist? You already know how to estimate limits with tables and graphs. Now you are going to calculate them exactly, without approximating.
Direct substitution and indeterminate forms
The first attempt is always to substitute $x=a$ into the function. If the function is a polynomial, a fraction whose denominator doesn't vanish, or a root with a positive radicand, the result of substituting is the limit.
This relies on the properties of limits, which let you find the limit of an operation from the limits of its parts.
But when you substitute, two kinds of results can appear that are not a number:
The temperature of an electronic component (in degrees Celsius) as a function of time $t$ (in hours) is given by the expression $T(t) = \frac{3t^2 - 5t + 12}{t + 2}$.…
The temperature of an electronic component (in degrees Celsius) as a function of time $t$ (in hours) is given by the expression $T(t) = \frac{3t^2 - 5t + 12}{t + 2}$. Calculate the limit of the temperature as $t$ approaches $3$ hours.
One-sided limits, piecewise functions, and parameters
When $x$ approaches $a$, it can do so from the left ($x\to a^-$) or from the right ($x\to a^+$). The limit exists only if both sides give the same value.
When you get $\frac{k}{0}$. You have to decide the sign on each side.
Piecewise functions. At the point where the formula changes, you calculate each one-sided limit with the formula that applies on that side. If both agree with the value of the function at that point, the function is continuous there.
Absolute value. $|x|=x$ if $x>0$ and $|x|=-x$ if $x<0$. That is why $\dfrac{|x|}{x}$ equals $1$ to the right of $0$ and $-1$ to the left.
Reading limits from a graph. Trace the curve with your finger toward the value of $x$ you are asked about, first from the left and then from the right, and pay attention to the height you reach, not to the filled or open dot that may be there. If the curve moves away upward or downward, the limit is $+\infty$ or $-\infty$. For $x\to+\infty$, look at where the right end settles.
Study the left- and right-hand limits of the function $f(x) = \frac{2x + 1}{x - 4}$ as $x$ tends to $4$ and determine whether the overall limit exists at that point.
Study the left- and right-hand limits of the function $f(x) = \frac{2x + 1}{x - 4}$ as $x$ tends to $4$ and determine whether the overall limit exists at that point.
The velocity of a test vehicle (in m/s) as a function of time $t$ (in seconds) is defined piecewise as $v(t) = 3t + 2$ if $0 \le t < 4$, and $v(t) = t^2 - 2$ if…
The velocity of a test vehicle (in m/s) as a function of time $t$ (in seconds) is defined piecewise as $v(t) = 3t + 2$ if $0 \le t < 4$, and $v(t) = t^2 - 2$ if $t \ge 4$. Determine using left- and right-hand limits whether the velocity function is continuous at time $t = 4$.
Given the following piecewise function, calculate the left- and right-hand limits at $x = 2$ and determine whether $\lim_{x \to 2} f(x)$ exists:…
Given the following piecewise function, calculate the left- and right-hand limits at $x = 2$ and determine whether $\lim_{x \to 2} f(x)$ exists:
$f(x) = \begin{cases} 2x + 1 & \text{if } x < 2 \\ x^2 + 1 & \text{if } x \ge 2 \end{cases}$
Study the existence of the limit at $x = 0$ of the function $f(x) = \frac{|x|}{x} + x^2$ by first calculating its left- and right-hand limits.
Study the existence of the limit at $x = 0$ of the function $f(x) = \frac{|x|}{x} + x^2$ by first calculating its left- and right-hand limits.
An electricity company sets its tariff $T(x)$ in pesos based on electricity consumption $x$ in kWh. For consumption less than $100$ kWh, the tariff is $T(x) = 3x + 100$.…
An electricity company sets its tariff $T(x)$ in pesos based on electricity consumption $x$ in kWh. For consumption less than $100$ kWh, the tariff is $T(x) = 3x + 100$. For consumption greater than or equal to $100$ kWh, the tariff changes to $T(x) = \frac{ax^2 + 4\,000}{x}$. Find the value of the parameter $a$ so that the limit of the tariff exists as consumption approaches $100$ kWh.
Observe the graph of the function $f(x)$ representing the temperature profile of a chemical process as a function of time $x$. Determine the following limits…
Observe the graph of the function $f(x)$ representing the temperature profile of a chemical process as a function of time $x$. Determine the following limits graphically: $\lim_{x \to 2^-} f(x)$, $\lim_{x \to 2^+} f(x)$ and $\lim_{x \to +\infty} f(x)$.
Consider a rectangle inscribed in a circle of radius $R = 10$ cm. If $x$ is the length of the base of the rectangle, its area is given by the function…
Consider a rectangle inscribed in a circle of radius $R = 10$ cm. If $x$ is the length of the base of the rectangle, its area is given by the function $A(x) = x \sqrt{400 - x^2}$. An architect needs to know how the ratio between the area of the rectangle and the square of its base behaves when the base becomes very small. Calculate $\lim_{x \to 0^+} \frac{A(x)}{x^2}$.
The relativistic mass of a moving particle is given by $m(v) = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $m_0$ is the rest mass, $v$ is the velocity of the particle…
The relativistic mass of a moving particle is given by $m(v) = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $m_0$ is the rest mass, $v$ is the velocity of the particle and $c$ is the speed of light. Calculate the limit of the mass as the velocity $v$ approaches the speed of light $c$ from the left ($v \to c^-$) and interpret the mathematical result in the physical context.
Factoring to resolve $\frac{0}{0}$
If the numerator and denominator are polynomials that both vanish at $x=a$, both have the factor $(x-a)$. You factor, cancel that factor, and then you can substitute.
This answers Ximena: $\dfrac{x^2-25}{x-5}=x+5$ for $x\neq5$, so the limit at $5$ equals $10$ even though the calculator shows an error.
When there is a subtraction of fractions, first combine them over a common denominator and then factor:
Calculate the following algebraic limit by resolving the indeterminate form using factorisation: $\lim_{x \to 5} \frac{x^2 - 25}{x^2 - 7x + 10}$
Calculate the following algebraic limit by resolving the indeterminate form using factorisation:
$\lim_{x \to 5} \frac{x^2 - 25}{x^2 - 7x + 10}$
Resolve the following indeterminate form by factorising the polynomials: $\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 5x + 6}$
Resolve the following indeterminate form by factorising the polynomials:
$\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 5x + 6}$
Calculate the value of the limit presenting an indeterminate form of the subtraction of algebraic fractions type:…
Calculate the value of the limit presenting an indeterminate form of the subtraction of algebraic fractions type:
$\lim_{x \to 2} \left( \frac{1}{x - 2} - \frac{4}{x^2 - 4} \right)$
The conjugate with roots
If the indeterminate form comes with a square root, multiply the numerator and denominator by the conjugate. Since $(a-b)(a+b)=a^2-b^2$, the root disappears from the place where it gets in the way.
With two roots in the numerator you conjugate the same way: $(\sqrt{A}-\sqrt{B})(\sqrt{A}+\sqrt{B})=A-B$.
With trigonometric functions you use the same trick and the known limit $\displaystyle\lim_{x\to0}\dfrac{\sin x}{x}=1$. For example, to find $\displaystyle\lim_{x\to0}\dfrac{1-\cos x}{x^2}$, multiply by $1+\cos x$: the numerator becomes $1-\cos^2x=\sin^2x$, and you get $\left(\dfrac{\sin x}{x}\right)^2\cdot\dfrac{1}{1+\cos x}\to 1\cdot\dfrac12$.
Evaluate the following limit by multiplying the numerator and denominator by the corresponding conjugate expression: $\lim_{x \to 3} \frac{\sqrt{x + 6} - 3}{x - 3}$
Evaluate the following limit by multiplying the numerator and denominator by the corresponding conjugate expression:
$\lim_{x \to 3} \frac{\sqrt{x + 6} - 3}{x - 3}$
Evaluate the following limit with indeterminate form $\frac{0}{0}$ containing two radicals in the numerator: $\lim_{x \to 0} \frac{\sqrt{4 + x} - \sqrt{4 - x}}{3x}$
Evaluate the following limit with indeterminate form $\frac{0}{0}$ containing two radicals in the numerator:
$\lim_{x \to 0} \frac{\sqrt{4 + x} - \sqrt{4 - x}}{3x}$
Given that $\lim_{x \to 0} \frac{\sin x}{x} = 1$, calculate the following trigonometric limit by making the necessary algebraic transformations (hint: multiply the…
Given that $\lim_{x \to 0} \frac{\sin x}{x} = 1$, calculate the following trigonometric limit by making the necessary algebraic transformations (hint: multiply the numerator and denominator by the conjugate of the numerator):
$\lim_{x \to 0} \frac{1 - \cos x}{x^2}$
Limits at infinity
When $x\to+\infty$, what matters is which term "dominates": the one with the highest power. The technique: divide the numerator and denominator by the highest power of $x$; the terms with $\frac{1}{x}$, $\frac{1}{x^2}$… tend to $0$.
With roots. For $x\to+\infty$, $\sqrt{9x^2+4x}$ behaves like $\sqrt{9x^2}=3x$; that is, the root "counts" as a term of degree $1$. When you divide by $x$ you put $x$ inside the root as $x^2$:
$$\frac{\sqrt{9x^2+4x}}{x}=\sqrt{9+\tfrac4x}\to3.$$Roots with different indices. If you subtract a cube root and a square root, the simple conjugate doesn't work. Compare each root separately with a linear expression that resembles it: for example, $\sqrt[3]{x^3+6x^2}$ resembles $x+2$ and $\sqrt{x^2+2x}$ resembles $x+1$. Add and subtract those expressions, resolve each difference (using $a^3-b^3=(a-b)(a^2+ab+b^2)$ for the cube root), and you will see that each difference tends to a number.
Parameters. If the limit at infinity must equal a number, set the dominant coefficients equal. For $\displaystyle\lim_{x\to+\infty}\dfrac{kx^3+\dots}{2x^3+\dots}=4$ you need $\frac{k}{2}=4$. If there is a subtraction $\infty-\infty$, first combine everything into a single fraction and require that the degree of the numerator not exceed that of the denominator.
Horizontal asymptote. If $\lim_{x\to+\infty}f(x)=L$, the graph approaches the line $y=L$. This is how you answer "what value does it tend toward…?" in a growth model with $e^{-0.2t}$: since $e^{-0.2t}\to0$ as $t\to+\infty$, the term $49e^{-0.2t}$ vanishes from the denominator.
Calculate the limit at infinity of the following function containing radicals, by analysing the degrees of the numerator and denominator:…
Calculate the limit at infinity of the following function containing radicals, by analysing the degrees of the numerator and denominator:
$\lim_{x \to +\infty} \frac{\sqrt{9x^2 + 4x} - 2x}{5x + 1}$
Calculate the value of the parameter $k$ so that the following limit equality holds: $\lim_{x \to +\infty} \frac{kx^3 - 5x + 1}{2x^3 + x^2 - 8} = 4$
Calculate the value of the parameter $k$ so that the following limit equality holds:
$\lim_{x \to +\infty} \frac{kx^3 - 5x + 1}{2x^3 + x^2 - 8} = 4$
The number of registered users on a new streaming platform, expressed in thousands, is modelled by the logistic function $U(t) = \frac{500}{1 + 49 e^{-0.2t}}$, where $t$…
The number of registered users on a new streaming platform, expressed in thousands, is modelled by the logistic function $U(t) = \frac{500}{1 + 49 e^{-0.2t}}$, where $t$ is the time in months since its launch. What value does the number of users tend towards over time?
Calculate the following limit, which presents an indeterminate form of type $\infty - \infty$ involving square roots:…
Calculate the following limit, which presents an indeterminate form of type $\infty - \infty$ involving square roots:
$\lim_{x \to +\infty} \left( \sqrt{x^2 + 6x} - x \right)$
Calculate the values of the parameters $a$ and $b$ so that the following equality holds at infinity:…
Calculate the values of the parameters $a$ and $b$ so that the following equality holds at infinity:
$\lim_{x \to +\infty} \left( \frac{ax^2 + bx - 1}{x + 2} - 3x \right) = 5$
Solve the following indeterminate form $\infty - \infty$ with radicals of different indices. Pay special attention to the algebraic strategy required to combine or treat…
Solve the following indeterminate form $\infty - \infty$ with radicals of different indices. Pay special attention to the algebraic strategy required to combine or treat the terms separately:
$\lim_{x \to +\infty} \left( \sqrt[3]{x^3 + 6x^2} - \sqrt{x^2 + 2x} \right)$
Extension: the number $e$ and exponentials
There are two situations here: the indeterminate form $1^\infty$, and $\frac{0}{0}$ with an exponential as $x\to0$.
The indeterminate form $1^\infty$
If the base tends to $1$ and the exponent tends to $\infty$, the indeterminate form is resolved with the number $e$.
Determine the value of the following limit by resolving the indeterminate form associated with the number $e$:…
Determine the value of the following limit by resolving the indeterminate form associated with the number $e$:
$\lim_{x \to +\infty} \left( \frac{x + 5}{x - 2} \right)^{3x}$
The $\frac{0}{0}$ with an exponential: $\dfrac{e^{ax}-1}{x}$
When $x\to0$, $e^{ax}\to1$, so $e^{ax}-1\to0$ and the quotient $\dfrac{e^{ax}-1}{x}$ is $\frac{0}{0}$. You can't factor or conjugate, but it is resolved with a supporting table and the idea of slope: $\dfrac{e^{ax}-e^{0}}{x-0}$ is the slope of the secant line between $x=0$ and $x$, and as $x$ approaches $0$ it becomes the slope of the tangent line to $e^{ax}$ at $0$.
If the limit you are asked for has an extra factor (for example, $x$ multiplying, or the $\frac12$ in an area), use the properties of limits: calculate each factor with what you already know and combine the results. If one factor tends to $0$ and the other to a number, the product tends to $0$.
In an optics experiment, a laser beam passes through a medium with a variable refractive index. The deflection of the beam defines a right-angled triangle whose base has…
In an optics experiment, a laser beam passes through a medium with a variable refractive index. The deflection of the beam defines a right-angled triangle whose base has length $x$ and whose height is given by the curve $h(x) = \frac{1 - e^{-2x}}{x}$. Look at the diagram and calculate the limit of the area of the shaded triangle $A(x)$ as $x \to 0^+$.