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 |
| 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 |
| None | 5) $\lim _{x \rightarrow \infty} \sqrt{x^{2}+2 x}… | Given the expression |
| None | $\lim _{x \rightarrow \infty}(1-\tanh x)^{\frac{1… | Rewrite the expression as: |
| 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 |
Suppose the function is |
| None |
Use the graph of |
The problem is asking for the limit of the function |
| None |
4. |
The given function is |
| None | L'Hôpital's rule does not help with the limit bel… | The given limit is |
| 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}… | |
| None | Find the limit \[ \lim _{x \rightarrow \infty}(1+… | We are given the limit |