Find the equations of any vertical asymptotes for the function below.

Ask by Lambert Moreno. in the United States

Mar 19,2025

Question

Find the equations of any vertical asymptotes for the function below. $$ f(x)=\frac{x^{2}+3 x-10}{x^{2}+2 x-15} $$

UpStudy AI · Accepted Answer

1. Factor the numerator and the denominator:  
   $$
   x^2 + 3x -10 = (x+5)(x-2)
   $$
   $$
   x^2 + 2x -15 = (x+5)(x-3)
   $$

2. Write the function with the factors:  
   $$
   f(x)=\frac{(x+5)(x-2)}{(x+5)(x-3)}
   $$

3. Cancel the common factor $$ (x+5) $$ with the note that cancellation is only valid when $$ x \neq -5 $$:  
   $$
   f(x)=\frac{x-2}{x-3}, \quad x\neq -5
   $$

4. Identify the vertical asymptotes by setting the denominator of the simplified expression equal to zero:  
   $$
   x-3=0 \quad \Rightarrow \quad x=3
   $$

5. Thus, the vertical asymptote is at  
   $$
   x=3.
   $$
The vertical asymptote is at $$ x = 3 $$.

Find the equations of any vertical asymptotes for the function below.

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