Finding Limits and Function Values from a Graph of $ f(x) $

Question

Given the graph of $ f(x) $, find $ \lim_{x \to -8} f(x) $, $ f(-8) $, $ \lim_{x \to -2} f(x) $, $ f(-2) $, $ \lim_{x \to 6} f(x) $, $ f(6) $, $ \lim_{x \to 10} f(x) $, $ f(10) $ (explain if DNE).

Accepted Answer

(a) $ \lim_{x \to -8} f(x) = -3 $; (b) $ f(-8) = -3 $; (c) $ \lim_{x \to -2} f(x) = 3 $; (d) $ f(-2) = 3 $; (e) $ \lim_{x \to 6} f(x) = 5 $; (f) $ f(6) = 5 $; (g) $ \lim_{x \to 10} f(x) = 0 $; (h) $ f(10) = 0 $

Answer

(a) $ \lim_{x \to -8} f(x) $: As $ x $ approaches $ -8 $ from both left and right, the graph approaches the dot at $ x = -8 $ (y - value $ -3 $), so the limit is $ -3 $. (b) $ f(-8) $: The value of $ f(x) $ at $ x = -8 $ is the y - value of the dot at $ x = -8 $, which is $ -3 $. (c) $ \lim_{x \to -2} f(x) $: As $ x $ approaches $ -2 $ from both left and right, the graph approaches the dot at $ x = -2 $ (y - value $ 3 $), so the limit is $ 3 $. (d) $ f(-2) $: The value of $ f(x) $ at $ x = -2 $ is the y - value of the dot at $ x = -2 $, which is $ 3 $. (e) $ \lim_{x \to 6} f(x) $: As $ x $ approaches $ 6 $ from both left and right, the graph approaches the closed dot at $ x = 6 $ (y - value $ 5 $), so the limit is $ 5 $. (f) $ f(6) $: The value of $ f(x) $ at $ x = 6 $ is the y - value of the closed dot at $ x = 6 $, which is $ 5 $. (g) $ \lim_{x \to 10} f(x) $: As $ x $ approaches $ 10 $ from the left, the graph approaches the closed dot at $ x = 10 $ (y - value $ 0 $). Since the right - hand limit (for $ x > 10 $) is not shown but the left - hand limit exists and is $ 0 $, the limit is $ 0 $. (h) $ f(10) $: The value of $ f(x) $ at $ x = 10 $ is the y - value of the closed dot at $ x = 10 $, which is $ 0 $.