Estimating Limits from Graphs

Finding the limit:

From the left: $\lim_{x \to 2^-} f(x) = 4$

From the right: $\lim_{x \to 2^+} f(x) = 4$

Since both one-sided limits exist and are equal:

$\lim_{x \to 2} f(x) = 4$

Comparing to f(2):

We're told that f(2) = 1.

Therefore: $\lim_{x \to 2} f(x) = 4 \neq f(2) = 1$

The limit does NOT equal the function value!

What this looks like:

• There's a hole at (2, 4) where the function "wants" to be
• There's a solid dot at (2, 1) where the function actually is
• This is called a removable discontinuity

Step 1: Understand what to look for: We need to see where the graph is heading as x approaches 2

Step 2: Trace from the left: As x moves toward 2 from the left, follow the curve to see what y-value it approaches

Step 3: Trace from the right: As x moves toward 2 from the right, follow the curve to see what y-value it approaches

Step 4: Since it's a smooth curve through (2, 3): Both left and right traces lead to y = 3

Step 5: Conclusion: lim(x→2) f(x) = 3

Note: In this case, the limit equals f(2), which is typical for continuous functions

Answer: 3

Step 1: Understand the graph: • Hole at (3, 5) means f(3) is undefined • But the curve approaches the point from both sides

Step 2: Check left-hand limit: As x → 3 from the left, the graph approaches y = 5

Step 3: Check right-hand limit: As x → 3 from the right, the graph also approaches y = 5

Step 4: Apply limit definition: Since both one-sided limits equal 5: lim(x→3) f(x) = 5

Step 5: Key insight: The limit exists even though f(3) is undefined! This is perfectly valid - limits describe nearby behavior

Answer: 5

Step 1: Identify one-sided limits: • lim(x→1⁻) f(x) = 2 (left side approaches 2) • lim(x→1⁺) f(x) = 4 (right side approaches 4)

Step 2: Check if they agree: Left-hand limit: 2 Right-hand limit: 4 2 ≠ 4

Step 3: Apply limit criterion: For lim(x→1) f(x) to exist, left and right limits must be equal

Step 4: Conclusion: Since they're different, the two-sided limit does not exist

Step 5: Proper notation: lim(x→1) f(x) = DNE (does not exist)

Answer: Does not exist (DNE)

Step 1: Identify the behavior: • Vertical asymptote at x = 2 • From the right (x > 2), graph increases without bound

Step 2: Interpret "increases without bound": The function values grow larger and larger They approach positive infinity

Step 3: Write the limit: lim(x→2⁺) f(x) = ∞

Step 4: Important notes: • This is an "infinite limit" • ∞ is not a real number • We often say "the limit does not exist" • But we can be more specific: it's unbounded positive

Step 5: Graphical meaning: Vertical asymptote where function shoots up to infinity

Answer: +∞ (infinite limit)