Asymptote Math Example 4

Follow the full solution, then compare it with the other examples linked below.

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For

f(x) = \dfrac{2x^2 - 3}{x^2 + 1}

, determine the horizontal asymptote and verify by computing

f(100)

1
Degrees are equal (both 2). Horizontal asymptote: ratio of leading coefficients $= \frac{2}{1} = 2$ , so $y = 2$ .
2
Verify: $f(100) = \frac{2(10000)-3}{10000+1} = \frac{19997}{10001} \approx 1.9995 \approx 2$ . ✓

Answer

Horizontal asymptote:

y = 2

x

grows large, lower-order terms become negligible, and the ratio approaches that of the leading terms. Numerical verification at

x=100

confirms the function is already very close to

2

About Asymptote

An asymptote is a line that a curve approaches arbitrarily closely as the input (or output) grows without bound, but typically never reaches.

Find the vertical and horizontal asymptotes of [formula].

Find the oblique (slant) asymptote of [formula].

Identify all asymptotes of [formula].