# Step-by-step Solution

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## Step-by-step explanation

Problem to solve:

$\lim_{x\to5}\left(\frac{x^2+8x-9}{x^3-x^2+5x-5}\right)$

Learn how to solve limits by direct substitution problems step by step online.

$1, 5$

Learn how to solve limits by direct substitution problems step by step online. Evaluate the limit of x^2+8x-9 as x approaches 5. We can factor the polynomial x^3-x^2+5x-5 using the rational root theorem, which guarantees that for a polynomial of the form a_nx^n+a_{n-1}x^{n-1}+\dots+a_0 there is a rational root of the form \pm\frac{p}{q}, where p belongs to the divisors of the constant term a_0, and q belongs to the divisors of the leading coefficient a_n. List all divisors p of the constant term a_0, which equals -5. Next, list all divisors of the leading coefficient a_n, which equals 1. The possible roots \pm\frac{p}{q} of the polynomial x^3-x^2+5x-5 will then be. Trying all possible roots, we found that 1 is a root of the polynomial. When we evaluate it in the polynomial, it gives us 0 as a result.

$\frac{7}{15}$$\,\,\left(\approx 0.4666666666666667\right)$
$\lim_{x\to5}\left(\frac{x^2+8x-9}{x^3-x^2+5x-5}\right)$