Limits of a Sequence (Leaving Cert Mathematics): Revision Notes

Limits of a Sequence

Introduction

Consider a bouncing a ball that is thrown at point $A$. The distance that the ball travels each time is halved. Will the ball ever reach the point $B$ ?

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The answer is no since the jump on each consecutive bounce gets smaller and smaller contributing less to the total distance of the ball.

We say that this sequence converges, meaning it has a finite limit. If a sequence has an infinite limit (or no limit), we say it diverges.

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• The sequence $1,3,5,7,...$ is divergent since the limit goes to infinity : $\lim_{n \to \infty} T_n=\infty$.
• The sequence $0.3, 0.33,0.333, 0.3333$ is convergent since $\lim_{n \to \infty } T_n = \tfrac{1}{3}$.
• The sequence $2,-2,2,-2,...$ is divergent because the limit does not exist.

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Example

Properties of Limits

Given that $c$ is a constant and $x$ is variable :

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Sum Rule

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$$\begin{align*} \lim_{x \to a}(f(x) \pm g(x))=\lim_{x \to a}f(x) \pm \lim_{x \to a}g(x) \end{align*}$$

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Constant Multiple Rule

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$$\begin{align*} \lim_{x \to a}cf(x)=c\lim_{x \to a}f(x) \end{align*}$$

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Product Rule

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$$\begin{align*} \lim_{x \to a}(f(x) \cdot g(x))=\lim_{x \to a}f(x) \cdot \lim_{x \to a}g(x) \end{align*}$$

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Quotient Rule

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$$\begin{align*} \lim_{x \to a}\left(\frac{f(x)}{g(x)}\right)=\frac{\lim_{a \to x}f(x)}{\lim_{a \to x}g(x)} \end{align*}$$

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Constant Rule

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$$\begin{align*} \lim_{x \to a}c=c \end{align*}$$

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Power Rule

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$$\begin{align*} \lim_{x \to a}(f(x))^x =\left(\lim_{x \to a}f(x) \right)^x \end{align*}$$

Composition Rule

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$$\begin{align*} \lim_{x \to a}f(g(x)) = f\left(\lim_{x \to a}g(x) \right ) \end{align*}$$

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Important Limits

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$$\begin{align*} \lim_{x \to \infty}\left( \frac{1}{x}\right) =0 \end{align*}$$

This should be intuitive as $x$ increases the denominator of the fraction while the numerator, stays constant, converging to $0$. Otherwise we could substitute in $\infty$, $\frac{1}{\infty}=0$.

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$$\lim_{x \to \infty}k^x \begin{cases} 0, & \text{if} -1<k<1 \\ \infty, & \text{if} -1>k,k>1\\ \end{cases}$$

For example, if $k=\frac{1}{2}$, then the limit of $\left( \frac{1}{2} \right)^x$ will converge to $0$ as $x$ increases to infinity.

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In the case of evaluating a limit that nears $\frac{\infty}{\infty}$, we have to take some extra steps since $\frac{\infty}{\infty}$ is not defined.

Example