Try This First
Try this short activity before reading the lesson.
> Take $f(x) = \dfrac{x^2 - 1}{x - 1}$. On paper, evaluate it at $x = 0.9$, $0.99$, $1.01$ and $1.1$. Then try $x = 1$. Write one sentence about what the four numbers are heading toward, and one about what happens at $1$ itself.
The Big Idea
A limit is where a function is heading, not where it is. $\frac{x^2-1}{x-1}$ has no value at $x = 1$, but every nearby value sits next to $2$. So $\lim_{x\to 1} \frac{x^2-1}{x-1} = 2$. The whole of calculus is built on this one separation: the value at a point, and the behavior near it.
Most limits on this topic fall to substitution. When substitution gives $\frac{0}{0}$, the expression is hiding a common factor: factor, or rationalize, and cancel it. When $x \to \infty$, only the fastest-growing terms matter.
Continuity is when the two things agree: the limit exists, $f(a)$ exists, and they are equal. Break any one of the three and you have a discontinuity, with a name for each kind.
Words to Know
- limit: $\lim_{x\to a} f(x) = L$ means $f(x)$ gets as close as you like to $L$ when $x$ is close enough to $a$. What $f(a)$ is, or whether it exists, is irrelevant.
- one-sided limit: $\lim_{x\to a^-}$ approaches from the left, $\lim_{x\to a^+}$ from the right. The two-sided limit exists exactly when both exist and agree.
- removable discontinuity: A hole: the limit exists, but $f(a)$ is missing or wrong. Redefine one point and the function is continuous. A jump has two different one-sided limits; an infinite discontinuity has a vertical asymptote.
- Intermediate Value Theorem: If $f$ is continuous on $[a,b]$, it takes every value between $f(a)$ and $f(b)$. Continuous is the hypothesis; without it, the theorem says nothing.
Formulas and Theorems
- Continuity at a point: $\lim_{x\to a} f(x) = f(a)$
- Note: Three things at once: the limit exists, $f(a)$ exists, and they agree.
- Squeeze theorem: $g(x) \le f(x) \le h(x),\ \lim g = \lim h = L \Rightarrow \lim f = L$
- Limit at infinity of a rational function: $\lim_{x\to\infty}\frac{a x^n + \dots}{b x^n + \dots} = \frac{a}{b}$
- Note: Same degree: the ratio of leading coefficients. Top degree lower: $0$. Higher: no finite limit.
Worked Examples and Your Turn
Study each example, then try the paired problem.
Example 1: Substitute first
Problem: Find $\displaystyle\lim_{x\to 2} \frac{x^2 + 3x}{x + 1}$.
Solution:
- Put $x = 2$ in: $\frac{4 + 6}{3} = \frac{10}{3}$. Why: a rational function is continuous wherever its bottom is not zero, and at a continuous point the limit is the value.
- Check the bottom: $2 + 1 = 3 \ne 0$. So substitution was legal and the answer stands.
Answer: $\dfrac{10}{3}$
> Notice: Always try substitution first. Only $\frac{0}{0}$ needs work. A nonzero number over $0$ is a different story (see the edges).
Your Turn 1: Find $\displaystyle\lim_{x\to -1} \frac{x^3 + 2}{x^2 + 4}$.
Example 2: The hole
Problem: Find $\displaystyle\lim_{x\to 3} \frac{x^2 - x - 6}{x - 3}$.
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 360 240" width="360" height="240" role="img" aria-label="Graph of y = (x² − x − 6)/(x − 3), which is the line y = x + 2 with one point missing. An open circle at (3, 5) marks the hole, and dashed guides show that the heights approach 5 as x approaches 3, so the limit is 5 even though the function is undefined at x = 3."><rect x="0" y="0" width="360" height="240" fill="#ffffff"/><line x1="138.0" y1="14.0" x2="138.0" y2="214.0" stroke="#c9c9c4" stroke-width="0.8" stroke-dasharray="3 3"/><line x1="179.6" y1="14.0" x2="179.6" y2="214.0" stroke="#c9c9c4" stroke-width="0.8" stroke-dasharray="3 3"/><line x1="221.2" y1="14.0" x2="221.2" y2="214.0" stroke="#c9c9c4" stroke-width="0.8" stroke-dasharray="3 3"/><line x1="262.8" y1="14.0" x2="262.8" y2="214.0" stroke="#c9c9c4" stroke-width="0.8" stroke-dasharray="3 3"/><line x1="304.4" y1="14.0" x2="304.4" y2="214.0" stroke="#c9c9c4" stroke-width="0.8" stroke-dasharray="3 3"/><line x1="34.0" y1="154.0" x2="346.0" y2="154.0" stroke="#c9c9c4" stroke-width="0.8" stroke-dasharray="3 3"/><line x1="34.0" y1="114.0" x2="346.0" y2="114.0" stroke="#c9c9c4" stroke-width="0.8" stroke-dasharray="3 3"/><line x1="34.0" y1="74.0" x2="346.0" y2="74.0" stroke="#c9c9c4" stroke-width="0.8" stroke-dasharray="3 3"/><line x1="34.0" y1="34.0" x2="346.0" y2="34.0" stroke="#c9c9c4" stroke-width="0.8" stroke-dasharray="3 3"/><line x1="34.0" y1="194.0" x2="346.0" y2="194.0" stroke="#333" stroke-width="1.5"/><line x1="96.4" y1="14.0" x2="96.4" y2="214.0" stroke="#333" stroke-width="1.5"/><text x="344.0" y="188.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="end">x</text><text x="102.4" y="24.0" font-family="Source Sans 3" font-size="12" fill="#333">y</text><line x1="138.0" y1="191.0" x2="138.0" y2="197.0" stroke="#333" stroke-width="1.5"/><text x="138.0" y="209.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle">1</text><line x1="179.6" y1="191.0" x2="179.6" y2="197.0" stroke="#333" stroke-width="1.5"/><text x="179.6" y="209.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle">2</text><line x1="221.2" y1="191.0" x2="221.2" y2="197.0" stroke="#333" stroke-width="1.5"/><text x="221.2" y="209.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle">3</text><line x1="262.8" y1="191.0" x2="262.8" y2="197.0" stroke="#333" stroke-width="1.5"/><text x="262.8" y="209.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle">4</text><line x1="304.4" y1="191.0" x2="304.4" y2="197.0" stroke="#333" stroke-width="1.5"/><text x="304.4" y="209.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="middle">5</text><line x1="93.4" y1="154.0" x2="99.4" y2="154.0" stroke="#333" stroke-width="1.5"/><text x="90.4" y="158.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="end">2</text><line x1="93.4" y1="114.0" x2="99.4" y2="114.0" stroke="#333" stroke-width="1.5"/><text x="90.4" y="118.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="end">4</text><line x1="93.4" y1="74.0" x2="99.4" y2="74.0" stroke="#333" stroke-width="1.5"/><text x="90.4" y="78.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="end">6</text><line x1="93.4" y1="34.0" x2="99.4" y2="34.0" stroke="#333" stroke-width="1.5"/><text x="90.4" y="38.0" font-family="Source Sans 3" font-size="12" fill="#333" text-anchor="end">8</text><polyline points="34.0,184.0 34.8,183.6 35.6,183.2 36.3,182.9 37.1,182.5 37.9,182.1 38.7,181.7 39.5,181.4 40.2,181.0 41.0,180.6 41.8,180.2 42.6,179.9 43.4,179.5 44.1,179.1 44.9,178.8 45.7,178.4 46.5,178.0 47.3,177.6 48.0,177.2 48.8,176.9 49.6,176.5 50.4,176.1 51.2,175.8 51.9,175.4 52.7,175.0 53.5,174.6 54.3,174.2 55.1,173.9 55.8,173.5 56.6,173.1 57.4,172.8 58.2,172.4 59.0,172.0 59.7,171.6 60.5,171.2 61.3,170.9 62.1,170.5 62.9,170.1 63.6,169.8 64.4,169.4 65.2,169.0 66.0,168.6 66.8,168.2 67.5,167.9 68.3,167.5 69.1,167.1 69.9,166.8 70.7,166.4 71.4,166.0 72.2,165.6 73.0,165.2 73.8,164.9 74.6,164.5 75.3,164.1 76.1,163.8 76.9,163.4 77.7,163.0 78.5,162.6 79.2,162.2 80.0,161.9 80.8,161.5 81.6,161.1 82.4,160.8 83.1,160.4 83.9,160.0 84.7,159.6 85.5,159.2 86.3,158.9 87.0,158.5 87.8,158.1 88.6,157.8 89.4,157.4 90.2,157.0 90.9,156.6 91.7,156.2 92.5,155.9 93.3,155.5 94.1,155.1 94.8,154.8 95.6,154.4 96.4,154.0 97.2,153.6 98.0,153.2 98.7,152.9 99.5,152.5 100.3,152.1 101.1,151.8 101.9,151.4 102.6,151.0 103.4,150.6 104.2,150.2 105.0,149.9 105.8,149.5 106.5,149.1 107.3,148.8 108.1,148.4 108.9,148.0 109.7,147.6 110.4,147.2 111.2,146.9 112.0,146.5 112.8,146.1 113.6,145.8 114.3,145.4 115.1,145.0 115.9,144.6 116.7,144.2 117.5,143.9 118.2,143.5 119.0,143.1 119.8,142.8 120.6,142.4 121.4,142.0 122.1,141.6 122.9,141.2 123.7,140.9 124.5,140.5 125.3,140.1 126.0,139.8 126.8,139.4 127.6,139.0 128.4,138.6 129.2,138.2 129.9,137.9 130.7,137.5 131.5,137.1 132.3,136.8 133.1,136.4 133.8,136.0 134.6,135.6 135.4,135.2 136.2,134.9 137.0,134.5 137.7,134.1 138.5,133.8 139.3,133.4 140.1,133.0 140.9,132.6 141.6,132.2 142.4,131.9 143.2,131.5 144.0,131.1 144.8,130.8 145.5,130.4 146.3,130.0 147.1,129.6 147.9,129.2 148.7,128.9 149.4,128.5 150.2,128.1 151.0,127.8 151.8,127.4 152.6,127.0 153.3,126.6 154.1,126.2 154.9,125.9 155.7,125.5 156.5,125.1 157.2,124.8 158.0,124.4 158.8,124.0 159.6,123.6 160.4,123.2 161.1,122.9 161.9,122.5 162.7,122.1 163.5,121.8 164.3,121.4 165.0,121.0 165.8,120.6 166.6,120.2 167.4,119.9 168.2,119.5 168.9,119.1 169.7,118.7 170.5,118.4 171.3,118.0 172.1,117.6 172.8,117.2 173.6,116.9 174.4,116.5 175.2,116.1 176.0,115.8 176.7,115.4 177.5,115.0 178.3,114.6 179.1,114.2 179.9,113.9 180.6,113.5 181.4,113.1 182.2,112.8 183.0,112.4 183.8,112.0 184.5,111.6 185.3,111.2 186.1,110.9 186.9,110.5 187.7,110.1 188.4,109.7 189.2,109.4 190.0,109.0 190.8,108.6 191.6,108.3 192.3,107.9 193.1,107.5 193.9,107.1 194.7,106.8 195.5,106.4 196.2,106.0 197.0,105.6 197.8,105.2 198.6,104.9 199.4,104.5 200.1,104.1 200.9,103.8 201.7,103.4 202.5,103.0 203.3,102.6 204.0,102.2 204.8,101.9 205.6,101.5 206.4,101.1 207.2,100.8 207.9,100.4 208.7,100.0 209.5,99.6 210.3,99.2 211.1,98.9 211.8,98.5 212.6,98.1 213.4,97.8 214.2,97.4 215.0,97.0 215.7,96.6 216.5,96.2 217.3,95.9 218.1,95.5 218.9,95.1 219.6,94.7 220.4,94.4 221.2,94.0 222.0,93.6 222.8,93.3 223.5,92.9 224.3,92.5 225.1,92.1 225.9,91.8 226.7,91.4 227.4,91.0 228.2,90.6 229.0,90.2 229.8,89.9 230.6,89.5 231.3,89.1 232.1,88.8 232.9,88.4 233.7,88.0 234.5,87.6 235.2,87.2 236.0,86.9 236.8,86.5 237.6,86.1 238.4,85.8 239.1,85.4 239.9,85.0 240.7,84.6 241.5,84.2 242.3,83.9 243.0,83.5 243.8,83.1 244.6,82.8 245.4,82.4 246.2,82.0 246.9,81.6 247.7,81.2 248.5,80.9 249.3,80.5 250.1,80.1 250.8,79.8 251.6,79.4 252.4,79.0 253.2,78.6 254.0,78.2 254.7,77.9 255.5,77.5 256.3,77.1 257.1,76.8 257.9,76.4 258.6,76.0 259.4,75.6 260.2,75.2 261.0,74.9 261.8,74.5 262.5,74.1 263.3,73.8 264.1,73.4 264.9,73.0 265.7,72.6 266.4,72.2 267.2,71.9 268.0,71.5 268.8,71.1 269.6,70.8 270.3,70.4 271.1,70.0 271.9,69.6 272.7,69.2 273.5,68.9 274.2,68.5 275.0,68.1 275.8,67.8 276.6,67.4 277.4,67.0 278.1,66.6 278.9,66.2 279.7,65.9 280.5,65.5 281.3,65.1 282.0,64.8 282.8,64.4 283.6,64.0 284.4,63.6 285.2,63.3 285.9,62.9 286.7,62.5 287.5,62.1 288.3,61.8 289.1,61.4 289.8,61.0 290.6,60.6 291.4,60.2 292.2,59.9 293.0,59.5 293.7,59.1 294.5,58.7 295.3,58.4 296.1,58.0 296.9,57.6 297.6,57.2 298.4,56.9 299.2,56.5 300.0,56.1 300.8,55.8 301.5,55.4 302.3,55.0 303.1,54.6 303.9,54.2 304.7,53.9 305.4,53.5 306.2,53.1 307.0,52.8 307.8,52.4 308.6,52.0 309.3,51.6 310.1,51.2 310.9,50.9 311.7,50.5 312.5,50.1 313.2,49.7 314.0,49.4 314.8,49.0 315.6,48.6 316.4,48.3 317.1,47.9 317.9,47.5 318.7,47.1 319.5,46.8 320.3,46.4 321.0,46.0 321.8,45.6 322.6,45.2 323.4,44.9 324.2,44.5 324.9,44.1 325.7,43.8 326.5,43.4 327.3,43.0 328.1,42.6 328.8,42.2 329.6,41.9 330.4,41.5 331.2,41.1 332.0,40.8 332.7,40.4 333.5,40.0 334.3,39.6 335.1,39.3 335.9,38.9 336.6,38.5 337.4,38.1 338.2,37.8 339.0,37.4 339.8,37.0 340.5,36.6 341.3,36.2 342.1,35.9 342.9,35.5 343.7,35.1 344.4,34.7 345.2,34.4 346.0,34.0" fill="none" stroke="#1f5f8b" stroke-width="2"/><line x1="221.2" y1="194.0" x2="221.2" y2="94.0" stroke="#333" stroke-width="1" stroke-dasharray="4 3"/><line x1="96.4" y1="94.0" x2="221.2" y2="94.0" stroke="#333" stroke-width="1" stroke-dasharray="4 3"/><circle cx="221.2" cy="94.0" r="4.2" fill="#ffffff" stroke="#1f5f8b" stroke-width="1.8"/><text x="231.2" y="110.0" font-family="Source Sans 3" font-size="12" fill="#333">hole at (3, 5)</text><text x="229.5" y="150.0" font-family="Source Sans 3" font-size="12" fill="#1f5f8b">y = (x² − x − 6)/(x − 3)</text><text x="229.5" y="170.0" font-family="Source Sans 3" font-size="12" fill="#1f5f8b">= x + 2 for x ≠ 3</text></svg>
Solution:
- Substitute: $\frac{9 - 3 - 6}{0} = \frac{0}{0}$. Why: this form is not an answer, it is an instruction. A zero on top and bottom means both share a factor of $(x - 3)$.
- Factor the top: $x^2 - x - 6 = (x - 3)(x + 2)$. Cancel with the bottom for every $x \ne 3$: the expression equals $x + 2$.
- Now let $x \to 3$: $3 + 2 = 5$. Why: the limit only cares about $x$ near $3$, and there the two expressions are identical.
Answer: $5$
> Notice: Say it in words: the hole is at $3$, the limit is $5$. The function is undefined at $3$. The limit is not.
> Verification & Check: $x = 3.01$: $\frac{9.0601 - 3.01 - 6}{0.01} = \frac{0.0501}{0.01} = 5.01$. Heading to $5$.
Your Turn 2: Find $\displaystyle\lim_{x\to -2} \frac{x^2 + 5x + 6}{x + 2}$.
Example 3: Rationalize
Problem: Find $\displaystyle\lim_{x\to 4} \frac{\sqrt{x} - 2}{x - 4}$.
Solution:
- Substitute: $\frac{0}{0}$. There is no visible factor to cancel because of the root. Why: the hidden factor appears when you multiply top and bottom by the conjugate $\sqrt{x} + 2$.
- $\dfrac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{(x - 4)(\sqrt{x} + 2)} = \dfrac{x - 4}{(x - 4)(\sqrt{x} + 2)} = \dfrac{1}{\sqrt{x} + 2}$.
- Let $x \to 4$: $\frac{1}{2 + 2} = \frac{1}{4}$.
Answer: $\dfrac{1}{4}$
> Notice: Same idea as the hole. The conjugate is just the tool that makes the shared factor visible when a root is in the way.
Your Turn 3: Find $\displaystyle\lim_{x\to 0} \frac{\sqrt{x + 1} - 1}{x}$.
Example 4: Far out
Problem: Find $\displaystyle\lim_{x\to\infty} \frac{5x^2 + x - 8}{2x^2 - 7}$.
The chapter continues in the book.
