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How do you use limits to find a horizontal asymptote?

Last updated date: 26th Feb 2024
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IVSAT 2024
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Hint: An asymptote can be expressed as the line on the graph of a function representing a value towards which the function may approach and it does not reach with certain exceptions. Horizontal asymptotes are the horizontal lines for which the graph of the function approaches as “x” tends to infinity. Here we will take an example to find the horizontal asymptote using limits.

Complete step-by-step solution:
Let us take an example,
$f(x) = \dfrac{{a{x^3} + b{x^2} + cx + d}}{{r{x^3} + s{x^2} + tx + u}}$
Apply the limit in the above expression which tends to infinity.
$\mathop {\lim }\limits_{x \to \infty } \dfrac{{a{x^3} + b{x^2} + cx + d}}{{r{x^3} + s{x^2} + tx + u}}$
Since, the power of “x” is cube therefore divide the above expression in the numerator and the denominator by common factors from the numerator and the denominator cancel each other. Also using the laws of power and exponent to simplify.
$\mathop {\lim }\limits_{x \to \infty } \dfrac{{a + \dfrac{b}{x} + \dfrac{c}{{{x^2}}} + \dfrac{d}{{{x^3}}}}}{{r + \dfrac{s}{x} + \dfrac{t}{{{x^2}}} + \dfrac{u}{{{x^3}}}}}$
Anything upon infinity gives resultant value as zero, when applied limit “x” and placing value as infinity.
$ = \dfrac{a}{r}$
And hence the required horizontal asymptote is $y = \dfrac{a}{r}$

Additional Information: Remember the seven basic rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through the below mentioned rules, it describes how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
-Product of powers rule
-Quotient of powers rule
-Power of a power rule
-Power of a product rule
-Power of a quotient rule
-Zero power rule
-Negative exponent rule

Note: Always remember that any number upon infinity always gives infinity whereas any number upon zero always gives us the value as the infinity. Be careful while placing the limits in the equation, first of all remove all the common factors from the equation. Always remember that the common factors from the numerator and the denominator cancel each other.