# Unit 2: One-Sided, Infinite Limits and Limits at Infinity (MT242) ## One-Sided Limits - **Right-hand limit** $\lim_{x\to c^+}f(x)=L$: for each $\varepsilon>0$ there is $\delta>0$ such that $cc$ (resp. $0$ such that $0<|x-c|<\delta$ implies $f(x)>\alpha$; similarly $-\infty$ with $f(x)<\alpha$. - It is a **divergence** to infinity, not a finite limit; the graph has a **vertical asymptote** $x=c$. - Examples: $\lim_{x\to0}\dfrac1{x^2}=+\infty$; $\lim_{x\to0^+}\dfrac1x=+\infty$ and $\lim_{x\to0^-}\dfrac1x=-\infty$. - **Related theorems (statements)**: - If $f\ge g$ near $c$ and $\lim g=+\infty$, then $\lim f=+\infty$. - If $\lim f=+\infty$ and $g$ is bounded below by $b>0$ (or has positive limit), then $\lim fg=+\infty$. - If $f>0$ then $\lim f=0\iff\lim\frac1f=+\infty$ (when $f\to0$ through positive values). ## Limits at Infinity - $\lim_{x\to\infty}f(x)=L$ if for each $\varepsilon>0$ there exists $K$ such that $x>K$ implies $|f(x)-L|<\varepsilon$; similarly $x\to-\infty$ with $x<-K$. The graph has a **horizontal asymptote** $y=L$. - Examples: $\lim_{x\to\infty}\dfrac1x=0$; $\lim_{x\to\infty}\dfrac{2x^2+1}{x^2+3}=2$; $\lim_{x\to\infty}\dfrac{\sin x}x=0$ (squeeze); $\lim_{x\to\infty}\left(1+\frac1x\right)^x=e$. - Rational functions: compare degrees: the limit is 0 if numerator degree is smaller, the ratio of leading coefficients if equal, and infinite if numerator degree is larger. - **Infinite limits at infinity**: $\lim_{x\to\infty}f(x)=\infty$ if for each $\alpha$ there is $K$ with $x>K\Rightarrow f(x)>\alpha$. For example $\lim x^n=\infty$, $\lim e^x=\infty$. - **Statements of related theorems**: the sum, product and quotient theorems hold; if $f\le g\le h$ for large $x$ and $f,h\to L$ then $g\to L$; if $\lim f/g=L>0$ (with $L$ finite) then $f\to\infty\iff g\to\infty$; and the **sequential criterion** applies with sequences $x_n\to\infty$. - **[Figure: Asymptotes]** Graph of $y=\frac{2x}{x-1}$ with vertical asymptote $x=1$ and horizontal asymptote $y=2$. ## Likely Exam Questions 1. Define the right-hand and left-hand limits. (2 marks) 2. Show that $\lim_{x\to0}\frac{|x|}x$ does not exist. (2 marks) 3. Define $\lim_{x\to\infty}f(x)=L$. (2 marks) 4. Prove that $\lim_{x\to c}f(x)$ exists iff both one-sided limits exist and are equal. (long) 5. Prove from the definition that $\lim_{x\to0}\frac1{x^2}=\infty$ and $\lim_{x\to\infty}\frac{x}{x+1}=1$. (long) 6. State the limit theorems for infinite limits and limits at infinity and use them to evaluate three limits. (long)