Limits

## Limits

Write $lim x → 2 f x = 10$ if $f\left(x\right)$ gets closer and closer to $10$ as $x$ gets closer and closer to $2$.

Example. Evaluate $\underset{x\to 2}{\mathrm{lim}}\frac{3{x}^{2}+8}{{x}^{2}-x}$.
When $x=1.5$, $\frac{3{x}^{2}+8}{{x}^{2}-x}=19.66\cdots$.
When $x=1.9$, $\frac{3{x}^{2}+8}{{x}^{2}-x}=11.011\cdots$.
When $x=1.99$, $\frac{3{x}^{2}+8}{{x}^{2}-x}=10.091\cdots$.
When $x=1.999$, $\frac{3{x}^{2}+8}{{x}^{2}-x}=10.00901\cdots$.
When $x=1.9999$, $\frac{3{x}^{2}+8}{{x}^{2}-x}=10.0009001\cdots$.

Usually determining the limit is straightforward.

Example. $\underset{x\to 1}{\mathrm{lim}}6{x}^{2}-4x+3=5$.

But sometimes …

Example. $\underset{x\to 1}{\mathrm{lim}}\frac{\sqrt{1+x}-1}{x}\stackrel{?}{=}\frac{0}{0}$.
Hoewver $\frac{0}{0}$ MAKES NO SENSE.

Example. $\underset{x\to 0}{\mathrm{lim}}\frac{5x}{x}\stackrel{?}{=}\frac{0}{0}$.
However here we have $\underset{x\to 0}{\mathrm{lim}}\frac{5x}{x}=\underset{x\to 0}{\mathrm{lim}}5=5$.

Example. $\underset{x\to 0}{\mathrm{lim}}\frac{17x}{2x}\stackrel{?}{=}\frac{0}{0}$.
However here we have $\underset{x\to 0}{\mathrm{lim}}\frac{17x}{2x}=\underset{x\to 0}{\mathrm{lim}}\frac{17}{2}=\frac{17}{2}$.

Example. $\underset{x\to 1}{\mathrm{lim}}\frac{\sqrt{1+x}-1}{x}\stackrel{?}{=}\frac{0}{0}$.
However here we have $lim x → 0 1 + x - 1 x = lim x → 0 1 + x - 1 x · 1 + x + 1 1 + x + 1 = lim x → 0 1 + x - 1 x 1 + x + 1 = lim x → 0 x x 1 + x + 1 = lim x → 0 1 1 + x + 1 = 1 1 + 0 + 1 = 1 2 .$ So whenever a limit looks like it is coming out to $\frac{0}{0}$ it needs to be looked at in a different way to see what it is really getting closer and closer to.

Example. Evaluate $\underset{x\to 7}{\mathrm{lim}}\frac{{x}^{2}-49}{x-7}$.
Here we have $lim x → 7 x 2 - 49 x - 7 = lim x → 7 x - 7 x + 7 x - 7 = lim x → 7 x + 7 = 7 + 7 = 14 .$

Example. Evaluate $\underset{x\to 5}{\mathrm{lim}}\frac{{x}^{5}-3125}{x-5}$.
Here we have $lim x → 5 x 5 - 3125 x - 5 = lim x → 5 x 5 - 5 5 x - 5 = lim x → 5 x - 5 x 4 + 5 x 3 + 5 2 x 2 + 5 3 x + 5 4 x - 5 = lim x → 5 x 4 + 5 x 3 + 5 2 x 2 + 5 3 x + 5 4 = 5 4 + 5 4 + 5 4 + 5 4 + 5 4 = 5 5 = 3125 .$

Example. Evaluate $\underset{x\to 5}{\mathrm{lim}}\frac{{x}^{5/2}-{a}^{5/2}}{x-a}$.
Here we have $lim x → 5 x 5 / 2 - a 5 / 2 x - a = lim x → 5 x 5 / 2 - a 5 / 2 x - a · x 5 / 2 + a 5 / 2 x 5 / 2 + a 5 / 2 = lim x → 5 x 5 - a 5 x - a · 1 x 5 / 2 + a 5 / 2 = lim x → 5 x - a x 4 + a x 3 + a 2 x 2 + a 3 x + a 4 x - a · 1 x 5 / 2 + a 5 / 2 = lim x → 5 x 4 + a x 3 + a 2 x 2 + a 3 x + a 4 x 5 / 2 + a 5 / 2 = a 4 + a 4 + a 4 + a 4 + a 4 a 5 / 2 + a 5 / 2 = 5 a 4 2 a 5 / 2 = 5 2 a 3 / 2 .$

## Particularly useful limits

Example. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}x}{x}$.
Here we have $lim x → 0 sin x x = lim x → 0 x - x 3 3 ! + x 5 5 ! - x 7 7 ! + x 9 9 ! - ⋯ x = lim x → 0 1 - x 2 3 ! + x 4 5 ! - x 6 7 ! + x 8 9 ! - ⋯ = 1 - 0 + 0 - 0 + 0 - ⋯ = 1$

Example. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{cos}x-1}{x}$.
Here we have $lim x → 0 cos x - 1 x = lim x → 0 1 - x 2 2 ! + x 4 4 ! - x 6 6 ! + x 8 8 ! - ⋯ - 1 x = lim x → 0 - x 2 2 ! + x 4 4 ! - x 6 6 ! + x 8 8 ! - ⋯ x = lim x → 0 - x 2 ! + x 3 4 ! - x 5 6 ! + x 7 8 ! - ⋯ = - 0 + 0 - 0 + 0 - ⋯ = 0 .$

Example. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{{e}^{x}-1}{x}$.
Here we have $lim x → 0 e x - 1 x = lim x → 0 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + ⋯ - 1 x = lim x → 0 x + x 2 2 ! + x 3 3 ! + x 4 4 ! + ⋯ x = lim x → 0 1 + x 2 ! + x 2 3 ! + x 3 4 ! + ⋯ = 1 + 0 + 0 + 0 + ⋯ = 1 .$

Example. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{ln}\left(1+x\right)}{x}$.
Let $y=\mathrm{ln}\left(1+x\right)$. Then So $lim x → 0 ln 1 + x x = lim y → 0 y e y - 1 = lim y → 0 1 e y - 1 y = 1 1 = 1 .$

Example. Evaluate $\underset{x\to 0}{\mathrm{lim}}{\left(1+x\right)}^{1/x}$.
We have $lim x → 0 1 + x 1 / x = lim x → 0 e ln 1 + x 1 / x = lim x → 0 e 1 x ln 1 + x = lim x → 0 e ln 1 + x x = e 1 = e .$

Note. $n\to \infty$ means as $n$ gets larger and larger.

Example. Evaluate $\underset{n\to \infty }{\mathrm{lim}}{\left(1+\frac{1}{n}\right)}^{n}$.
Let $x=\frac{1}{n}$. Then $x\to 0$ as $n\to \infty$. So $lim n → ∞ 1 + 1 n n = lim x → 0 1 + x 1 / x = e .$.

Example. Evaluate $\underset{x\to \pi }{\mathrm{lim}}\frac{\mathrm{sin}x}{x-\pi }$.
Let $y=x-\pi$. Then $y\to 0$ as $x\to \pi$. So $lim x → π sin x x - π = lim y → 0 sin y + π y = lim y → 0 sin y cos π + cos y sin π y = lim y → 0 sin y -1 + cos y 0 y = lim y → 0 - sin y y = 1$

Example. Evaluate $\underset{x\to \infty }{\mathrm{lim}}\frac{{x}^{2}-7x+11}{3{x}^{2}+10}$.
We have $lim x → ∞ x 2 - 7 x + 11 3 x 2 + 10 = lim x → ∞ 1 - 7 x + 11 x 2 3 + 10 x 2 = 1 - 0 + 0 3 + 0 = 1 3 .$

Example. Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}3x}{\mathrm{sin}5x}$.
We have $lim x → 0 sin 3 x sin 5 x = lim x → 0 sin 3 x 3 x · 3 x · 5 x sin 5 x · 1 5 x = lim x → 0 sin 3 x 3 x · 1 sin 5 x 5 x · 3 x 5 x = 1 · 1 1 · 3 5 = 3 5 .$

Example. Evaluate $\underset{x\to 1}{\mathrm{lim}}\frac{1-x}{{\left(\mathrm{arccos}x\right)}^{2}}$.
Let $y=\mathrm{arccos}x$. Then $y\to 0$ as $x\to 1$ and $x=\mathrm{cos}y$. So $lim x → 1 1 - x arccos x 2 = lim y → 0 1 - cos y y 2 = lim y → 0 1 - cos y y 2 · 1 + cos y 1 + cos y = lim y → 0 1 - cos 2 y y 2 · 1 1 + cos y = lim y → 0 sin y y · sin y y · 1 1 + cos y = 1 · 1 · 1 2 = 1 2 .$

Example. Evaluate $\underset{\Delta x\to 0}{\mathrm{lim}}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}$ when $f\left(x\right)=\mathrm{sin}2x$.
We have $lim Δ x → 0 f x + Δ x - f x Δ x = lim Δ x → 0 sin 2 x + Δ x - sin 2 x Δ x = lim Δ x → 0 sin 2 x + 2 Δ x - sin 2 x Δ x = lim Δ x → 0 sin 2 x cos 2 Δ x + cos 2 x sin 2 Δ x - sin 2 x Δ x = lim Δ x → 0 sin 2 x · cos 2 Δ x - 1 Δ x + cos 2 x · sin 2 Δ x Δ x = lim Δ x → 0 sin 2 x · cos 2 Δ x - 1 2 Δ x · 2 + cos 2 x · sin 2 Δ x 2 Δ x · 2 = sin 2 x · 0 · 2 + cos 2 x · 1 · 2 = 2 cos 2 x .$

Example. Evaluate $\underset{\Delta x\to 0}{\mathrm{lim}}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}$ when $f\left(x\right)=\mathrm{cos}{x}^{2}$.
We have $lim Δ x → 0 f x + Δ x - f x Δ x = lim Δ x → 0 cos x + Δ x 2 - cos x 2 Δ x = lim Δ x → 0 cos x 2 + 2 x Δ x + Δ x 2 - cos x 2 Δ x = lim Δ x → 0 cos x 2 cos 2 x Δ x + Δ x 2 - sin x 2 sin 2 x Δ x + Δ x 2 - cos x 2 Δ x = lim Δ x → 0 cos x 2 · cos 2 x Δ x + Δ x 2 - 1 Δ x - sin x 2 · sin 2 x Δ x + Δ x 2 Δ x = lim Δ x → 0 cos x 2 · cos 2 x Δ x + Δ x 2 - 1 2 x Δ x + Δ x 2 · 2 x Δ x + Δ x 2 Δ x - sin x 2 · sin 2 x Δ x + Δ x 2 2 x Δ x + Δ x 2 · 2 x Δ x + Δ x 2 Δ x = lim Δ x → 0 cos x 2 · cos 2 x Δ x + Δ x 2 - 1 2 x Δ x + Δ x 2 · 2 x + Δ x - sin x 2 · sin 2 x Δ x + Δ x 2 2 x Δ x + Δ x 2 · 2 x + Δ x = cos x 2 · 0 · 2 x - sin x 2 · 1 · 2 x = - 2 x sin x 2 ,$ (since $2x\Delta x+{\left(\Delta x\right)}^{2}\to 0$ as $\Delta x\to 0$).

Example. Evaluate $\underset{\Delta x\to 0}{\mathrm{lim}}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}$ when $f\left(x\right)={x}^{x}$.
We have $lim Δ x → 0 f x + Δ x - f x Δ x = lim Δ x → 0 x + Δ x x + Δ x + x x Δ x = lim Δ x → 0 e ln x + Δ x x + Δ x + e ln x x Δ x = lim Δ x → 0 e x + Δ x ln x + Δ x + e x ln x Δ x = lim Δ x → 0 e x ln x · e x + Δ x ln x + Δ x - x ln x + 1 Δ x = lim Δ x → 0 e x ln x · e x + Δ x ln x + Δ x - x ln x + 1 x + Δ x ln x + Δ x - x ln x · x + Δ x ln x + Δ x - x ln x Δ x = lim Δ x → 0 e x ln x · e x + Δ x ln x + Δ x - x ln x + 1 x + Δ x ln x + Δ x - x ln x x ln x + Δ x - x ln x Δ x + ln x + Δ x = lim Δ x → 0 e x ln x · e x + Δ x ln x + Δ x - x ln x + 1 x + Δ x ln x + Δ x - x ln x x ln x 1 + Δ x x - x ln x Δ x + ln x + Δ x = lim Δ x → 0 e x ln x · e x + Δ x ln x + Δ x - x ln x + 1 x + Δ x ln x + Δ x - x ln x x ln x + ln 1 + Δ x x - x ln x Δ x + ln x + Δ x = lim Δ x → 0 e x ln x · e x + Δ x ln x + Δ x - x ln x + 1 x + Δ x ln x + Δ x - x ln x x ln 1 + Δ x x Δ x + ln x + Δ x = lim Δ x → 0 e x ln x · e x + Δ x ln x + Δ x - x ln x + 1 x + Δ x ln x + Δ x - x ln x ln 1 + Δ x x Δ x x + ln x + Δ x = e x ln x · 1 · 1 + ln x = x x + x x ln x .$ (since $\left(x+\Delta x\right)\mathrm{ln}\left(x+\Delta x\right)-x\mathrm{ln}x\to 0$ and $\Delta x/x\to 0$ as $\Delta x\to 0$).