Limits at a point
You already know what it means for a function to have a limit at a point. Now it is time to calculate it precisely in specific cases: sometimes substituting is enough, sometimes you have to factorise, and sometimes it is worth using a table before tackling the algebra.
Direct substitution
If the function is defined and "well-behaved" (a polynomial, a rational function whose denominator does not vanish, a root with a positive argument...) at the point, the limit is calculated simply by substituting.
Calculate the following limits by direct substitution into the expression of the function: a) $\lim_{x \to -1} (x^3 - 3x^2 + 4)$ b) $\lim_{x \to 0} \frac{2x + 5}{x - 3}$…
Calculate the following limits by direct substitution into the expression of the function:
- a) $\lim_{x \to -1} (x^3 - 3x^2 + 4)$
- b) $\lim_{x \to 0} \frac{2x + 5}{x - 3}$
- c) $\lim_{x \to 4} \sqrt{x + 5}$
A market study indicates that the daily profits of a new coffee shop, in euros, follow the function $B(t) = t^2 + 5t - 10$, where $t$ is the time in months since its…
A market study indicates that the daily profits of a new coffee shop, in euros, follow the function $B(t) = t^2 + 5t - 10$, where $t$ is the time in months since its opening. Calculate the limit of the profits as time tends to $3$ months ($t \to 3$).
Estimating limits with a table of values
Before calculating algebraically, a table of values very close to the point — from both sides — lets you guess where the function is heading. It is especially useful when direct substitution gives $\frac00$.
Given the function $f(x) = \frac{x^2 - 1}{x - 1}$, complete the following table of values to approximate the limit of the function as $x$ tends to $1$. What value does…
Given the function $f(x) = \frac{x^2 - 1}{x - 1}$, complete the following table of values to approximate the limit of the function as $x$ tends to $1$. What value does the function appear to approach?
| $x \to 1^-$ | $f(x)$ | $x \to 1^+$ | $f(x)$ |
|---|---|---|---|
| $0,9$ | $1,1$ | ||
| $0,99$ | $1,01$ | ||
| $0,999$ | $1,001$ |
A content creator has observed that the growth of followers on their channel during the first hour of a livestream follows the function $f(x) = \frac{x^2 - 25}{x - 5}$,…
A content creator has observed that the growth of followers on their channel during the first hour of a livestream follows the function $f(x) = \frac{x^2 - 25}{x - 5}$, where $x$ is the time in minutes. However, right at minute $x = 5$, the platform suffered a small technical error and did not record data.
Estimate the value that the number of followers approaches as time approaches $5$ minutes by completing the following table of values and calculating the limit:
| $x$ | $4,9$ | $4,99$ | $4,999$ | $5,001$ | $5,01$ | $5,1$ |
|---|---|---|---|---|---|---|
| $f(x)$ |
$\lim_{x \to 5} \frac{x^2 - 25}{x - 5}$
The indeterminate form $\frac00$: factorise and simplify
When the numerator and denominator are polynomials that vanish at the same point $a$, both have the factor $(x-a)$. By factorising and cancelling that factor, the indeterminate form disappears.
Given the piecewise-defined function: $$f(x) = \begin{cases} 2x + 1 & \text{if } x < 1 \\ x^2 + 2 & \text{if } x \geq 1 \end{cases}$$ Calculate the one-sided limits at…
Given the piecewise-defined function:
$$f(x) = \begin{cases} 2x + 1 & \text{if } x < 1 \\ x^2 + 2 & \text{if } x \geq 1 \end{cases}$$Calculate the one-sided limits at the boundary point:
- a) $\lim_{x \to 1^-} f(x)$
- b) $\lim_{x \to 1^+} f(x)$
Does the limit $\lim_{x \to 1} f(x)$ exist? Justify your answer.
Calculate the value of the following limit by resolving the indeterminate form by factorising polynomials: $\lim_{x \to -2} \frac{x^3 + 3x^2 + 2x}{x^2 - x - 6}$
Calculate the value of the following limit by resolving the indeterminate form by factorising polynomials:
$\lim_{x \to -2} \frac{x^3 + 3x^2 + 2x}{x^2 - x - 6}$
The conjugate with roots
If the indeterminate form $\frac00$ appears together with a root, multiplying and dividing by the conjugate turns the root into a square and removes the problem.
Solve the following limit using the conjugate product method to eliminate the radical from the numerator: $\lim_{x \to 4} \frac{\sqrt{x+5} - 3}{x - 4}$
Solve the following limit using the conjugate product method to eliminate the radical from the numerator:
$\lim_{x \to 4} \frac{\sqrt{x+5} - 3}{x - 4}$
Vertical asymptote or removable discontinuity?
When the denominator vanishes at $x=a$, everything depends on whether the numerator also vanishes there: if it does, the common root cancels and there may be no asymptote; if it does not, there is a genuine vertical asymptote with infinite limits.
Given the rational function $f(x) = \frac{2x^2 - 8}{x^2 - 4x + 4}$, study its behaviour as $x$ approaches $2$. To do so, calculate the one-sided limits:…
Given the rational function $f(x) = \frac{2x^2 - 8}{x^2 - 4x + 4}$, study its behaviour as $x$ approaches $2$.
To do so, calculate the one-sided limits:
$\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$
Explain, based on the results, whether the function has a vertical asymptote or a removable discontinuity at that point.
In a biology laboratory, the concentration of a toxin in a bacterial culture is studied. The concentration $C$ in parts per million (ppm) with respect to the temperature…
In a biology laboratory, the concentration of a toxin in a bacterial culture is studied. The concentration $C$ in parts per million (ppm) with respect to the temperature $x$ in degrees Celsius follows the function:
$C(x) = \frac{x^2 - 4x + 3}{x^2 - 1}$
Calculate the limiting concentration as the temperature tends to $1^\circ\text{C}$. Is the concentration defined at exactly that temperature? Justify your answer by analysing the domain and the limit at that point.
Consider the rational function $R(x) = \frac{x^2 - m}{x^2 - 5x + 6}$. a) Determine the value of $m$ so that the function has a removable discontinuity at $x = 2$. b) For…
Consider the rational function $R(x) = \frac{x^2 - m}{x^2 - 5x + 6}$.
- a) Determine the value of $m$ so that the function has a removable discontinuity at $x = 2$.
- b) For that value of $m$, calculate $\lim_{x \to 2} R(x)$.
- c) What happens to the limit of the function at $x = 3$ for the value of $m$ found in part a)? Calculate the one-sided limits at $x = 3$ and describe the behavior of the infinite branches of the function near that line.