Determine the oblique asymptote of the graph of the function.

Ask by Murray Washington. in the United States

Mar 25,2025

Question

Determine the oblique asymptote of the graph of the function. $$ g(x)=\frac{x^{2}+3 x-7}{x+1} $$

UpStudy AI · Accepted Answer

1. We start with the function
   $$
   g(x)=\frac{x^{2}+3x-7}{x+1}.
   $$

2. Since the degree of the numerator is one higher than the degree of the denominator, there is an oblique asymptote. We find it by performing polynomial long division.

3. Divide $$ x^2+3x-7 $$ by $$ x+1 $$:

- **First term:** $$ \frac{x^2}{x} = x $$. Multiply $$ x $$ by $$ x+1 $$ to get $$ x^2+x $$. Subtract from the numerator:
     $$
     (x^2+3x-7) - (x^2+x) = 2x-7.
     $$
     
   - **Next term:** $$ \frac{2x}{x} = 2 $$. Multiply $$ 2 $$ by $$ x+1 $$ to get $$ 2x+2 $$. Subtract:
     $$
     (2x-7) - (2x+2) = -9.
     $$

The division yields:
   $$
   g(x)=x+2-\frac{9}{x+1}.
   $$

4. As $$ x $$ tends to infinity or negative infinity, the term $$-\frac{9}{x+1}$$ approaches $$0$$. Hence, the oblique asymptote is given by the quotient:
   $$
   y=x+2.
   $$

Thus, the oblique asymptote of the graph of $$ g(x) $$ is $$ y=x+2 $$.
The oblique asymptote of the graph of $$ g(x) = \frac{x^{2}+3x-7}{x+1} $$ is $$ y = x + 2 $$.

Determine the oblique asymptote of the graph of the function.

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