When we talk about assessing limits that approach infinity, we're discussing the process of identifying the value a function gravitates towards as the input (x) starts to lean towards infinity. This method is essential in comprehending how a function behaves when the inputs are extremely large. Several techniques such as direct substitution, factoring, rationalizing, or employing L'Hopital's Rule for indeterminate forms, can be utilized in this process.
| Topic | Problem | Solution |
|---|---|---|
| None | Find the limit of the following sequence or deter… | The sequence is given by the formula \(\frac{4n}{\sqrt{49n^2+6}}\). To find the limit of this seque… |
| None | (b) $\lim _{x \rightarrow \infty} \tan ^{-1}\left… | First, we need to understand the meaning of the question. The question is asking for the limit as x… |
| None | Find the limit of the sequence, using L'Hôpital's… | We are given the sequence \(\frac{n^{2}}{8^{n}}\) and asked to find its limit as n approaches infin… |
| None | 5) $\lim _{x \rightarrow \infty} \sqrt{x^{2}+2 x}… | Given the expression \(\sqrt{x^{2}+2 x}-\sqrt{x^{2}-8 x}\), we are asked to find the limit as x app… |
| None | $\lim _{x \rightarrow \infty}(1-\tanh x)^{\frac{1… | Rewrite the expression as: \(\lim _{x \rightarrow \infty}e^{\frac{1}{x}\ln(1-\tanh x)}\) |
| None | \( \lim _{n \rightarrow \infty} \frac{n^{2}\left(… | \( \lim _{n \rightarrow \infty} \frac{n^{2}\left(\sin n+\cos ^{3} n\right)}{\left(n^{2}+1\right)(n-… |
| None | a. Suppose $f(x)=\frac{1}{x}$. As $x \rightarrow … | Suppose the function is \(f(x)=\frac{1}{x}\). |
| None | Use the graph of $h(x)=x^{3}-4 x$ to find the fol… | The problem is asking for the limit of the function \(h(x)=x^{3}-4 x\) as \(x\) approaches \(\infty… |
| None | 4. $[0 / 1$ Points] DETAILS PREVIOUS ANSWERS SCAL… | The given function is \(f = \sqrt{\frac{49x^{3} + 5x - 9}{x^{3} - 6x + 4}}\). |
| None | L'Hôpital's rule does not help with the limit bel… | The given limit is \(\lim _{x \rightarrow \infty} \frac{\sqrt{25 x+5}}{\sqrt{x+5}}\). |
| None | (3) $\lim _{x \rightarrow-\infty} \frac{2^{x+1}}{… | Rewrite the function as a single exponential function: \(\frac{2^{x+1}}{3^x} = \frac{2^x \cdot 2}{3… |
| None | \( \lim _{n \rightarrow \infty} \frac{\sqrt{a^{2}… | \( \lim _{n \rightarrow \infty} \frac{\sqrt{a^{2}+n}-\sqrt{a^{2}-n}}{n} \) = \( \lim _{n \rightarro… |
| None | Find the limit \[ \lim _{x \rightarrow \infty}(1+… | We are given the limit \(\lim _{x \rightarrow \infty}(1+2 x)^{\frac{3}{2 \ln x}}\) |