In 1st Bachillerato you already learned to compare the degrees of polynomials and rational functions when $x\to\pm\infty$. Now the challenge is subtler: comparing infinities of different kinds (an exponential against a polynomial, a logarithm against a root) and resolving indeterminate forms that can no longer be sorted out just by looking at degrees.
Revision: polynomials and rational functions, now with parameters
In university entrance exams (EBAU) these rules almost always come with an unknown parameter that you have to find from a condition on the limit.
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Determine the value of the following limits based on comparing the degrees of the polynomials: a) $\lim_{x \to +\infty} \frac{5x^2 - 3x + 8}{2x^2 + 1}$ b)…
Determine the value of the following limits based on comparing the degrees of the polynomials:
Given the rational function $f(x) = \dfrac{ax^2 + 3x - 5}{2x^2 - k}$, calculate the values of the real parameters $a$ and $k$ knowing that the function has a horizontal…
Given the rational function $f(x) = \dfrac{ax^2 + 3x - 5}{2x^2 - k}$, calculate the values of the real parameters $a$ and $k$ knowing that the function has a horizontal asymptote at $y = 3$ and that its graph passes through the point $(1, -2)$. Once found, check whether the limit as $x \to -\infty$ coincides with the limit as $x \to +\infty$.
Hierarchy of infinities: exponential, polynomial and logarithm
When $x\to+\infty$, not all "infinities" grow at the same rate. There is a fixed order worth memorising: if you divide the one that grows faster by the one that grows more slowly, the limit is $+\infty$; the other way round, it is $0$.
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Given the functions $f(x) = 2^x$ and $g(x) = x^{10} + 500x^2$, reason which of the two functions grows faster as $x \to +\infty$ using the scale of infinities. From your…
Given the functions $f(x) = 2^x$ and $g(x) = x^{10} + 500x^2$, reason which of the two functions grows faster as $x \to +\infty$ using the scale of infinities. From your answer, deduce the value of the following limit:
Given the functions $f(x) = e^x$, $g(x) = x^{10}$, and $h(x) = \ln(x)$, order the following limits from smallest to largest according to the order of magnitude of the…
Given the functions $f(x) = e^x$, $g(x) = x^{10}$, and $h(x) = \ln(x)$, order the following limits from smallest to largest according to the order of magnitude of the infinities involved as $x \to +\infty$:
The number $e$ as a limit: the $1^\infty$ indeterminate form
There is a special limit that appears again and again when studying growth (compound interest, populations, decay): a base that tends to $1$ raised to an exponent that tends to $\infty$. This form, $1^\infty$, is indeterminate and is resolved using the number $e$.
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Complete the following table of values for the function $f(x) = \left( \frac{x+1}{x} \right)^x$ and use the results obtained to conjecture which famous number the…
Complete the following table of values for the function $f(x) = \left( \frac{x+1}{x} \right)^x$ and use the results obtained to conjecture which famous number the function tends to as $x$ becomes very large.
Determine the value of the following limit using its relation to the number $e$. You must transform the expression so that it takes the form of the definition of this…
Determine the value of the following limit using its relation to the number $e$. You must transform the expression so that it takes the form of the definition of this constant:
Roots: the conjugate technique for $\infty-\infty$
When you get a subtraction of roots that individually both tend to $+\infty$ or both tend to $-\infty$, you cannot simply subtract: you have to multiply and divide by the conjugate to turn the subtraction into a quotient.
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Solve the following limit involving a difference of irrational functions. Remember to use the conjugate method to resolve the indeterminate form $[\infty - \infty]$:…
Solve the following limit involving a difference of irrational functions. Remember to use the conjugate method to resolve the indeterminate form $[\infty - \infty]$:
Study the limit at negative infinity of the following rational function containing an absolute value and a square root: $$f(x) = \frac{|3x - 2| + \sqrt{x^2 + 5x}}{2x -…
Study the limit at negative infinity of the following rational function containing an absolute value and a square root:
That is, calculate $\lim_{x \to -\infty} f(x)$. Take special care with the sign of the variable $x$ when inserting or extracting factors from the square root and when removing the absolute value.
Calculate the limit of the sequence of functions given by the following nested root: $$\lim_{x \to +\infty} \left( \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x} \right)$$…
Calculate the limit of the sequence of functions given by the following nested root:
Back to the start: To compare a logarithm against a root such as $\lim_{x\to+\infty}\frac{\ln(x)}{\sqrt{x}}$, write the root as the power $x^{0{,}5}$; by the scale of infinities ($n=0{,}5>0$), the power outgrows the logarithm and the limit is $0$.
Summary
Let's practice! 10 medium exercises in this section