Question

question 6
evaluate the limit:
\\(\lim_{y\to 7} \frac{y^3 - 343}{y - 7}\\) =
question help: \\(\square\\) video
submit question

Explanation:

Step1: Recognize the indeterminate form

Substitute $y=7$: $\frac{7^3 - 343}{7-7} = \frac{0}{0}$, so factor the numerator.

Step2: Factor the difference of cubes

Use $a^3 - b^3 = (a-b)(a^2+ab+b^2)$. Here $a=y$, $b=7$, so:
$y^3 - 343 = (y-7)(y^2 +7y + 49)$

Step3: Simplify the rational function

Cancel the common factor $(y-7)$:
$\lim_{y \to 7} \frac{(y-7)(y^2 +7y + 49)}{y-7} = \lim_{y \to 7} (y^2 +7y + 49)$

Step4: Evaluate the simplified limit

Substitute $y=7$:
$7^2 +7(7) + 49 = 49 + 49 + 49$

Answer:

147