Consider the following expression.
\[ \lim_{x \to -3} \frac{\sqrt{2x + 22} - 4}{x + 3} \] The value of the above expression (rounded to 2 decimal places) is \(\underline{\hspace{2cm}}\).

Question

Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( |f(x) - f(y)| \leq (x - y)^2 \) for all \( x, y \in \mathbb{R} \). Then \[ f(1) - f(0) = \ \underline{\hspace{2cm}} \quad \text{(Answer in integer)} \]

Answer 1

Step 1: Observe the indeterminate form.
Direct substitution of x = −3 gives the form 0 / 0, so we simplify the expression.

Step 2: Rationalize the numerator.

\[ \frac{\sqrt{2x+22} - 4}{x+3} \times \frac{\sqrt{2x+22} + 4}{\sqrt{2x+22} + 4} \]

\[ = \frac{(2x+22) - 16}{(x+3)(\sqrt{2x+22} + 4)} \]

\[ = \frac{2x + 6}{(x+3)(\sqrt{2x+22} + 4)} \]

\[ = \frac{2(x+3)}{(x+3)(\sqrt{2x+22} + 4)} \]

Step 3: Cancel the common factor.

\[ = \frac{2}{\sqrt{2x+22} + 4} \]

Step 4: Substitute x = −3.

\[ = \frac{2}{\sqrt{16} + 4} = \frac{2}{8} = 0.25 \]

Final Answer:

0.25

Consider the following expression.
\[ \lim_{x \to -3} \frac{\sqrt{2x + 22} - 4}{x + 3} \] The value of the above expression (rounded to 2 decimal places) is \(\underline{\hspace{2cm}}\).

Show Hint

Correct Answer: 0.25

Solution and Explanation

Top Questions on Limit, continuity and differentiability

Questions Asked in GATE CS exam

Consider the following expression. \[ \lim_{x \to -3} \frac{\sqrt{2x + 22} - 4}{x + 3} \] The value of the above expression (rounded to 2 decimal places) is \(\underline{\hspace{2cm}}\).

Show Hint

Correct Answer: 0.25

Solution and Explanation

Top Questions on Limit, continuity and differentiability

Questions Asked in GATE CS exam

Consider the following expression.
\[ \lim_{x \to -3} \frac{\sqrt{2x + 22} - 4}{x + 3} \] The value of the above expression (rounded to 2 decimal places) is \(\underline{\hspace{2cm}}\).