Limits, question 24

Topic: Limits
Difficulty: Challenging
Suggested time: 6 min
Marks: 4
Theory: UBC Math 100 §2.1 CLP-1 §1.3Key results

Determine both one-sided limits below, and state with justification whether the two-sided limit $\lim_{x \to 2} \dfrac{x - 5}{x - 2}$ exists.

\lim_{x \to 2^{-}} \frac{x - 5}{x - 2} \qquad \text{and} \qquad \lim_{x \to 2^{+}} \frac{x - 5}{x - 2}

Write your reasoning in full, then compare it against the model solution.

Your answer

Theory UBC Math 100 §2.1 CLP-1 §1.3Key results

Worked solution

Strategy

A non-zero numerator over a denominator approaching zero means the magnitude grows without bound. The only real work is tracking the sign from each side, so handle the two one-sided limits separately.

Working

$x \to 2 \implies x - 5 \to -3$
The numerator approaches a fixed negative number from both sides.
$x \to 2^{-} \implies x - 2 \to 0^{-}$
Just below 2, say x = 1.9, the denominator is small and negative.
$\lim_{x \to 2^{-}} \frac{x - 5}{x - 2} = +\infty$
The numerator tends to −3 and the denominator to 0 through negative values. A negative number over a small negative one is large and positive, so the quotient grows without bound.
$x \to 2^{+} \implies x - 2 \to 0^{+}$
Just above 2, say x = 2.1, the denominator is small and positive.
$\lim_{x \to 2^{+}} \frac{x - 5}{x - 2} = -\infty$
Now the denominator tends to 0 through positive values. A negative number over a small positive one is large and negative, so the quotient falls without bound.
The two one-sided limits disagree, so the two-sided limit does not exist. The line x = 2 is a vertical asymptote, with the graph rising on the left and falling on the right.

Answer

\lim_{x \to 2^{-}} \frac{x - 5}{x - 2} = +\infty, \quad \lim_{x \to 2^{+}} \frac{x - 5}{x - 2} = -\infty, \quad \lim_{x \to 2} \frac{x - 5}{x - 2} \text{ does not exist}

What earns the marks

States that the numerator tends to −3, a non-zero value, while the denominator tends to 0.
Determines the sign of the denominator on each side of 2 separately.
Concludes +∞ from the left and −∞ from the right.
States that the two-sided limit does not exist, and identifies x = 2 as a vertical asymptote.

Check

At x = 1.99 the quotient is −3.01/(−0.01) = 301 and at x = 2.01 it is −2.99/0.01 = −299: huge, with opposite signs.

Where this goes wrong

Answering “∞” or “undefined” without separating the two sides. Marks here are for the sign analysis, and the two sides genuinely disagree, so a single unsigned answer cannot be right.

Leans on

Sign analysis near a root, from Functions, graphs and inverses

Try a similar question

Evaluate the one-sided limit.

\lim_{x \to 4^{+}} \frac{x + 2}{4 - x}

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Limits practice question 24