Limits in indeterminate forms (l’Hopital’s rule): notes and practice questions

(a) Prove by mathematical induction that $\frac{d^n}{dx^n}(xe^{-x}) = (-1)^n (x-n)e^{-x}$ for $n \in \mathbb{Z}^+$.

(b) Hence or otherwise, determine the Maclaurin series of $f(x) = xe^{-x}$ in ascending powers of $x$, up to and including the term in $x^4$.

(c) Hence or otherwise, determine the value of $\lim_{x\to0} \frac{(xe^{-x} - x)^2}{x^4}$.

The function $f$ is defined by $f(x) = e^x \sinh x$, where $x \in \mathbb{R}$.

Find the Maclaurin series for $f(x)$ up to and including the $x^3$ term.

Hence, find an approximate value for $\int_0^1 e^{x^2} \sinh(x^2)dx$.

The function $g$ is defined by $g(x) = e^x \cosh x$, where $x \in \mathbb{R}$.

Show that $g''(x) = 2g'(x)$.

Hence, find the values of $g'''(0)$ and $g^{(4)}(0)$.

Using the result from part (c), find the Maclaurin series for $g(x)$ up to and including the $x^4$ term.

Hence, or otherwise, determine the value of $\lim_{x \to 0} \frac{2e^x \cosh x - 2 - 2x - 2x^2}{x^3}$.

(a) Prove by mathematical induction that $\frac{d^n}{dx^n}(xe^{-x}) = (-1)^n(x-n)e^{-x}$ for $n \in \mathbb{Z}^+$.

(b) Hence or otherwise, find the Maclaurin series of $f(x) = xe^{-x}$ in ascending powers of $x$, up to and including the term in $x^5$.

(c) Hence or otherwise, determine the value of $\lim_{x\to0} \frac{xe^{-x} - x + x^2}{x^3}$.

(a) Prove by mathematical induction that $\frac{d^n}{dx^n}\left(\frac{x}{1-x}\right) = n!(1-x)^{-(n+1)}$ for $n \in \mathbb{Z}^+$.

(b) Hence or otherwise, determine the Maclaurin series of $f(x) = \frac{x}{1-x}$ in ascending powers of $x$, up to and including the term in $x^4$.

(c) Hence or otherwise, determine the value of $\lim_{x\to0} \frac{\left(\frac{x}{1-x} - x\right)^2}{x^4}$.

Consider the function $g(x)=\frac{1-\cos(ax)}{x^2}$, where $x \neq 0$ and $a \in \mathbb{R}^+$.

(a) Show that $g$ is an even function.

(b) Given that $\lim_{x\to0} g(x) = 8$, find the value of $a$.

Consider the function $g(x)=\frac{1-\cos(ax)}{e^{x^2}-1}$, where $x \neq 0$ and $a \in \mathbb{R}^+$.

(a) Show that $g$ is an even function.

(b) Given that $\lim_{x\to0} g(x) = 8$, find the value of $a$.

Consider the family of integrals defined by $I_n = \int x^n e^{-x} \, dx$ for $n \in \mathbb{N}_0$, where $\mathbb{N}_0 = \{0, 1, 2, ...\}$.

(a) By using integration by parts, show that $I_n = -x^n e^{-x} + n I_{n-1}$ for $n \ge 1$.

(b) Hence, find an explicit expression for $\int x^3 e^{-x} \, dx$.

(c) The region $R$ is enclosed by the graph of $y = x^{3/2} e^{-x/2}$ and the $x$-axis for $x \ge 0$. The region $R$ is rotated by $2\pi$ radians about the $x$-axis. Find the volume of the solid generated.

(d) Show that $\lim_{x \to \infty} x^n e^{-x} = 0$ for any $n \in \mathbb{N}$.

Consider the function $h(x) = x e^{-x}$.

(e) (i) Find the Maclaurin series for $h(x)$ up to and including the term in $x^4$.

(ii) Hence, find the value of the fourth derivative of $h(x)$ at $x=0$, i.e. $h^{(4)}(0)$.

Consider the limit $\lim_{x\to0} \frac{\ln(\cos x + 1) -k}{x^2}$, where $k \in \mathbb{R}$.

Show that a finite limit only exists for $k = \ln 2$.

Using l'Hôpital's rule, show algebraically that the value of the limit is $-\frac{1}{4}$.

Find the following limits, if they exist.

(a) $\lim_{x \to -1} (x^3 - 2x^2 + x - 5)$

(b) $\lim_{x \to 4} \frac{2x - 1}{x^2 - 3}$

(c) $\lim_{x \to 0} \sqrt{9 - 4x^2}$

(d) $\lim_{x \to -2} \frac{x^2 + 5x + 6}{x+2}$

(e) $\lim_{x \to -\infty} \frac{3x}{\sqrt{4x^2 + 1}}$

(f) $\lim_{x \to \infty} \frac{5x^2 - 3}{x^3 + 2x}$

Consider the differential equation $\frac{dy}{dx} = y^2 \sin x$.

Given that $y = 1$ when $x = 0$, show that the solution to the equation is $y = \sec x$.

Determine the value of the constant $A$ for which the following limit exists, and evaluate the limit:

$\lim_{x\to 0} \frac{\sec x - A}{x^2}$

A chemical engineer is studying the concentration of a certain byproduct in a reaction mixture over time. The concentration is modelled by the function $C_1(t) = t e^{-2t}$, where $t \ge 0$ represents time in hours.

(a) Sketch the graph of $y = C_1(t)$, stating the coordinates of the maximum concentration.

(b) The total amount of byproduct accumulated in the first $b$ hours is given by the integral $\int_0^b C_1(t) dt$.

Show that $\int_0^b t e^{-2t} dt = \frac{1}{4} (1 - (2b+1)e^{-2b})$.

(c.i) The total amount of byproduct accumulated indefinitely, $A_1$, is given by $\lim_{b \to \infty} \int_0^b C_1(t) dt$.

Use l'Hôpital's rule to find $\lim_{b \to \infty} \frac{1 - (2b+1)e^{-2b}}{4}$. You may assume that the condition for applying l'Hôpital's rule has been met.

(c.ii) Hence write down the value of $A_1$.

(d) The total amount of byproduct accumulated indefinitely for a general function $C_n(t) = t^n e^{-2t}$ is denoted by $A_n = \int_0^\infty t^n e^{-2t} dt$.

You are given that $A_2 = 0.25$ and $A_3 = 0.375$.

(i) Use your graphic display calculator, and an appropriate value for the upper limit, to determine the value of $A_4$.

(ii) Determine the value of $A_5$.

(e) Suggest an expression for $A_n$ in terms of $n$, where $n \in \mathbb{Z}^+$.

(f) Use mathematical induction to prove your conjecture from part (e). You may assume that, for any value of $m$, $\lim_{t \to \infty} t^m e^{-2t} = 0$.