Evaluate the following limits.

1. \( \displaystyle \lim_{x \rightarrow 5 } \frac{x^2-2x-15}{x^2-25} \)

Solution

2. \( \displaystyle \lim_{x \rightarrow 1} \frac{x^2-3x+2}{\ln{x}} \)

Solution

3. \( \displaystyle \lim_{x \rightarrow 0} \frac{3\sin{4x}}{5x} \)

Solution

4. \( \displaystyle \lim_{x \rightarrow \infty } \frac{3x^3}{e^x} \)

Solution

5. \( \displaystyle \lim_{x \rightarrow 0 } \frac{1-\cos{2x}}{2x^2}\)

Solution

6. \( \displaystyle \lim_{x \rightarrow \frac{\pi}{2}^- } \frac{1+\tan{x}}{\sec x} \)

Solution

7. \( \displaystyle \lim_{x \rightarrow -\infty } x^3\;e^x \)

Solution

8. \( \displaystyle \lim_{x \rightarrow 0^+ } \sqrt{x} \; \ln{x} \)

Solution

9. \( \displaystyle \lim_{x \rightarrow \infty} x^2 \; \sin \left(\frac{1}{4x^2}\right) \)

Solution