Could you do 6 and 7 and use definition of limits? Thanks ### Limits of Functions and SequencesThis section is based on the mathematical concepts of limits, specifically focusing on whether certain functions have limits at specified points. For practical understanding, examples are included along with exercises that require proofs and justifications.#### Exercises and Examples:6. **Define \( f : (0, 1) \to \mathbb{R} \) by \( f(x) = \cos \frac{1}{x} \).** - **Task:** Does \( f \) have a limit at 0? Justify.7. **Define \( f : (0, 1) \to \mathbb{R} \) by \( f(x) = x \cos \frac{1}{x} \).** - **Task:** Does \( f \) have a limit at 0? Justify.8. **Define \( f : (0, 1) \to \mathbb{R} \) by \( f(x) = \frac{x^3 - x^2 + x - 1}{x - 1} \).** - **Task:** Prove that \( f \) has a limit at 1.9. **Define \( f : (-1, 1) \to \mathbb{R} \) by \( f(x) = \frac{x + 1}{x^2 - 1} \).** - **Task:** Does \( f \) have a limit at 1? Justify.#### Key Concepts:- **Limits of Functions and Sequences:** 10. Consider \( f : (0, 2) \to \mathbb{R} \) defined by \( f(x) = x^t \). - **Task:** Assume that \( f \) has a limit at 0. - **Hint:** Choose a sequence \( \{x_n\}_{n=1}^{\infty} \) converging to 0 such that the limit of \( \{f(x_n)\} \) is easy to determine.11. Suppose \( f, g, \) and \( h : D \to \mathbb{R} \) where \( x_0 \) is an accumulation point of \( D \), \( f(x) = g(x) \) for all \( x \in D \), and \( f \) and \( h \

As per guidelines we are supposed to answer only one question. In this case question 6 answered.Here we use the limit theorem. If two sequences converge at the same point but cos of those converges to different points then limit of cos does not exist.Limit of cos(1/x) does not exists.